DuPont Quality Management and Technology SOE/MTB - 1.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Section 1
INTRODUCTION
DuPont Quality Management and Technology SOE/MTB - 1.2© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHAT IS DESIGN OF EXPERIMENTS (DOE)?
DOE is a strategic process, with supporting methods and tools, for guiding the
- planning- execution- analysis of results- application of results
of experimental or developmental programs.
DuPont Quality Management and Technology SOE/MTB - 1.3© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
BENEFITS OF DOE
Highest leverage quality tool available --- use to design quality in up front
Reduces product/process development cycle time Most efficient strategy for gaining process
understanding Develops true cause-and-effect relationships Provides solution to current problem plus
information for solving future problems An objective, fact-based system for decision
making, complete with quantitative measures of uncertainty
DuPont Quality Management and Technology SOE/MTB - 1.4© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHERE CAN DOE BE APPLIED?
DOE is useful in every stage of product life cycle– Product and Process Development– Process Scale-Up, Operations Startup, Customer
Verification– Process Control, Product and Process
Improvement
DuPont Quality Management and Technology SOE/MTB - 1.5© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
APPLICATIONS IN PRODUCT ANDPROCESS DEVELOPMENT
Define and translate customer needs Design robust products Design robust processes Reduce time to commercialization Develop test methods Define operating procedures Enhance business integration of R&D with
Operations
DuPont Quality Management and Technology SOE/MTB - 1.6© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
APPLICATIONS IN SCALE-UP, STARTUPAND CUSTOMER VERIFICATION
Reduce time to process qualification Identify key process variables Determine product specifications Design product field tests Develop standard operating conditions and
standard operating procedures
DuPont Quality Management and Technology SOE/MTB - 1.7© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
APPLICATIONS IN PROCESS CONTROL
Develop process models Calibrate process control knobs Adjust to changing customer needs Troubleshoot process problems Develop sampling protocols
DuPont Quality Management and Technology SOE/MTB - 1.8© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
APPLICATIONS IN PRODUCT AND PROCESS IMPROVEMENT
Reduce product variability Improve first-pass first-quality yield Increase capacity Reduce transition time Make process more robust Improve test methods
DuPont Quality Management and Technology SOE/MTB - 1.9© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHAT YOU SHOULD GET OUT OF SOE
An appreciation of the basic underlying concepts of DOE
A general strategy for approaching experimentation
A set of efficient, widely-applicable tools for designing and analyzing experiments
Hands-on experience at using these tools Some sense of when and where to seek
expert help
DuPont Quality Management and Technology SOE/ECHIP - 2.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Section 2
WORKSHOP 1
DuPont Quality Management and Technology SOE/ECHIP - 2.2© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WORKSHOP 1 PROBLEM
A new polymer is being readied for a plant process. One major problem remains: the COLOR (yellowness) has often been unacceptable in experimental production to date. The COLOR value should be made as low as possible. Prior work has indicated that COLOR may be affected by the following variables.
EXP. RANGEABBREV. VARIABLE NAME LOW HIGH UNITS
C Catalyst Concentration 1.0 1.8 %T Reactor Temperature 130 190 deg CA Additive Amount 1 5 kg
Also from prior work, it is known that MODULUS can be predicted by the following equation over the experimental range of interest:
MODULUS = -69.5 + 100*C + 0.15*T - 5.0*A
DuPont Quality Management and Technology SOE/ECHIP - 2.3© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Your problem is:1. To demonstrate an approximate set of conditions
to obtain low COLOR together with low MODULUS, and
2. To support your conclusion with a description of the effects of the factors on the response, COLOR.
Your boss’s best guess of a good place to start is:CATALYST = 1.25 %TEMPERATURE = 137 deg CADDITIVE = 3 kg
WORKSHOP 1 PROBLEM
DuPont Quality Management and Technology SOE/ECHIP - 2.4© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WORKSHOP 1 TEAM REPORTS
TeamRec. Settings
Cata Temp Addi Color Mod.# ofRuns
123456789
10
ResultsCata Temp Addi
Effects on ColorMethod/Comments
DuPont Quality Management and Technology SOE/ECHIP - 2.5© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WORKSHOP 1 TEAM REPORTS
TeamRec. Settings
Cata Temp Addi Color Mod.# ofRuns
11121314151617181920
ResultsCata Temp Addi
Effects on ColorMethod/Comments
DuPont Quality Management and Technology SOE/ECHIP - 2.6© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WORKSHOP 1:Concepts Introduced
Experimental Variability Is a Fact of Life Properties Can Be Represented As
Functions of Control Variables Geometry of Experimental Region Contour Plots Interaction Multiple Responses
DuPont Quality Management and Technology SOE/ECHIP - 3.1© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Section 3
FOUNDATIONS OF THE STRATEGY
DuPont Quality Management and Technology SOE/ECHIP - 3.2© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
UNDERLYING PRINCIPLES
World is multivariate Experimental error is a fact of life Experimentation is a process Multi-stage approach Statistical strategy The 6 B’s of DOE
DuPont Quality Management and Technology SOE/ECHIP - 3.3© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
WORLD IS MULTIVARIATE
Almost always more than one factor of interest that can be varied
e.g. pressure, temperature, pH, flow rate, hold time, screw speed
Interactions often present --- factor effects not additive i.e. synergistic or antagonistic effects
Usually, several responses (outcomes) of intereste.g. viscosity, yield, assay, color, hardness, modulus, dyeability
Tradeoffs between responses often necessary
DuPont Quality Management and Technology SOE/ECHIP - 3.4© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
VARIABLES
RESPONSESFACTORS LURKINGVARIABLES
variables that aredeliberately controlled
in the experiment
outcome variablesthat are measured
during the experiment
variables that areunidentified oruncontrolled
synonymsDEPENDENT VARIABLES
PROPERTIESCHARACTERISTICS
OUTCOMES
INDEPENDENT VARIABLESPREDICTORS
KNOBSPROCESS VARIABLES
TREATMENTSDOSES
NOISEUNCONTROLLED VARIABLES
DuPont Quality Management and Technology SOE/ECHIP - 3.5© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
EXPERIMENTAL ERROR
Experimental error is the “noise” in the system --- the catchall term used to explain why results are not identical from replicate to replicate
2 types of experimental error:– systematic or bias error– random error
DuPont Quality Management and Technology SOE/ECHIP - 3.6© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
RANDOM ERROR
0 0
•Unpatterned variability
•Unpredictable
•Multiple unassignable causes
•Normal error distribution
•Standard deviation represents “typical” error
DuPont Quality Management and Technology SOE/ECHIP - 3.7© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
SYSTEMATIC OR BIAS ERROR
•Patterned variation
•May be predictable
•Due to single assignable cause–e.g. shift, raw material lot, day, tool wear, ambient temperature
DuPont Quality Management and Technology SOE/ECHIP - 3.8© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
DEALING WITH EXPERIMENTAL ERROR
EXPERIMENTAL ERROR
RANDOM ERROR SYSTEMATIC ERROR
CAUSE Unassignable Assignable
NATURE Unpatterned Patterned
REMEDY Replication BlockingRandomization
DuPont Quality Management and Technology SOE/ECHIP - 3.9© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
REPLICATION
true relationship
fitted relationship
random error
random error
true relationship
fitted relationship
average
average
•Through averaging of replicates the impact of random error is reduced
•Two forms of replication–Hidden replication --- feature of all good designs
–Pure replication (as above)
FACTOR FACTOR
RES
PON
SE
RES
PON
SE
DuPont Quality Management and Technology SOE/ECHIP - 3.10© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
BLOCKING
Used in presence of identifiable source of potential bias (“blocking factor”)
– e.g. day, raw material lot Split the experiment up into blocks
representing different levels of the blocking factor
Keep the blocks balanced with respect to the experimental factors. This prevents confounding, or confusing, any block effect with that of an experimental factor.
DuPont Quality Management and Technology SOE/ECHIP - 3.11© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
RANDOMIZATION
“Insurance” protection against potential unidentified sources of bias
Randomize the order of experimental runs If experiment is blocked, randomize within
blocks May require constrained randomization if
some experimental factors hard to change
DuPont Quality Management and Technology SOE/ECHIP - 3.12© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
EXPERIMENTATION IS A PROCESS
Gather Information
Define Objectives
Design Experiment
Run Experiment
Analyze Experiment
Interpret Results
Perform Confirmation Runs
Go tonext stage of
experimentation?
Apply Results
Update Information
yes
no
DuPont Quality Management and Technology SOE/ECHIP - 3.13© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
MULTI-STAGE APPROACH
SCREENING INTERACTION RESPONSEDESIGNS DESIGNS SURFACE
DESIGNS
Evolution of the Experimental Environment
NUMBER OF 6 or more 3 - 8 2 - 6FACTORS
OBJECTIVE Identify key factors Understand factor Prediction modelinteractions Optimization
COMMON Plackett-Burman Full Factorial Box-BehnkenDESIGNS Fractional Factorial Fractional Factorial Central Composite
(resolution 3 or 4) (resolution 5) Face Center Cube
DuPont Quality Management and Technology SOE/ECHIP - 3.14© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
STATISTICAL STRATEGYVS. ONE-FACTOR-AT-A-TIME
Y
X1
Y
X1
Levelsof X2
Levelsof X2
One-Factor-At-A-Time Statistical
DuPont Quality Management and Technology SOE/ECHIP - 3.15© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
COMPARING THE STRATEGIES
1-FACTOR-AT-A-TIME STATISTICAL
DESIGN
FITTED MODEL
EXPERIMENTAL ERROR
INTERACTIONS
Vary only 1 factor at time,in multiple small increments,keeping all others fixed
Curves fitted through datapoints, separately for each factor
Ignored
Not considered; not estimable
Vary all factors jointly in balanced bite-sized space-filling design
Simple empirical models based on low-order polynomials
Recognized and estimated
Dealt with as appropriate
DuPont Quality Management and Technology SOE/ECHIP - 3.16© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
6 B’S OF DOE
Bite Size– Just enough runs to meet objectives, achieve desired sensitivity, and
estimate experimental error Boldness
– Vary experimental factors over wide range– Measure all relevant responses
Balance– Use balanced designs to maximize efficiency and minimize confounding
Bias Error– Take countermeasures such as randomization and blocking
Blunders– Avoid through careful planning and execution
Batting Average– Improve your odds of success through statistical designs and empirical
modeling
DuPont Quality Management and Technology SOE/ECHIP - 4.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Section 4
FACTORIAL GEOMETRY
DuPont Quality Management and Technology SOE/ECHIP - 4.2© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
EVOLUTION OF THE ENVIRONMENT:Intermediate Stage
SCREENING RESPONSEDESIGNS SURFACE
DESIGNS
Evolution of the Experimental Environment
NUMBER OF 6 or more 3 - 8 2 - 6FACTORS
OBJECTIVE Identify key factors Understand factor Prediction modelinteractions Optimization
COMMON Plackett-Burman Full Factorial Box-BehnkenDESIGNS Fractional Factorial Fractional Factorial Central Composite
(resolution 3 or 4) (resolution 5) Face Center Cube
INTERACTIONDESIGNS
DuPont Quality Management and Technology SOE/ECHIP - 4.3© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
TWO-LEVEL FACTORIAL DESIGNS
2k Distinct Runs Easy to Plan and Analyze Usable for Either Continuous or Discrete
Factors with Two Levels Uniformly Spread Through Factor Space Permit Estimation of Both Main Factor
Effects and Interaction Effects
DuPont Quality Management and Technology SOE/ECHIP - 4.4© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
23 FACTORIAL DESIGN
X3
X2
X1
(HI, HI, HI)
(HI, LO, HI)
(LO, HI, HI)
(LO, LO , HI)
(HI, HI, LO)
(HI, LO, LO)
(LO, HI, LO)
(X1, X2, X3) =(LO, LO, LO)
DuPont Quality Management and Technology SOE/ECHIP - 4.5© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
3 FACTOR 2 LEVEL FULL-FACTORIALExperiment Data
CATALYST TEMPERATURE ADDITIVE COLOR MODULUS1.81.01.81.01.81.81.01.0
190190130190130190130130
51551151
7248374836746261
11454
10534
1251342545
DuPont Quality Management and Technology SOE/ECHIP - 4.6© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
23 FACTORIAL DESIGNwith Workshop 1 COLOR Data
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
DuPont Quality Management and Technology SOE/ECHIP - 4.7© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHAT IS AN EFFECT ?
An effect is the difference in the averagesof two groups of observations.
Let YLOW = average of values on LOW plane.Let YHIGH = average of values on HIGH plane.
EFFECT = YHIGH - YLOW
DuPont Quality Management and Technology SOE/ECHIP - 4.8© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: EFFECT OF X1
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
72 + 37 + 74 + 36 48 + 62 + 48 + 614 4
= 24 + (-25) + 26 + (-25)4
= 0
24
-25
26
-25
DuPont Quality Management and Technology SOE/ECHIP - 4.9© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: EFFECT OF X2
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
48 + 72 + 48 + 74 62 + 37 + 61 + 364 4
= -14 + 35 + (-13) + 384
= 11.5
-14
-13
35
38
DuPont Quality Management and Technology SOE/ECHIP - 4.10© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: EFFECT OF X3
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
48 + 72 + 62 + 37 48 + 74 + 61 + 364 4
= 0 + (-2) + 1 + 14 = 0
0 -2
1 1
DuPont Quality Management and Technology SOE/ECHIP - 4.11© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
ONE FACTOR AT A TIME
X3
X2
X1
No Hidden ReplicationNot Space Filling
DuPont Quality Management and Technology SOE/ECHIP - 4.12© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
INTERACTION OF X1 AND X2
Y
X1
Low X2
High X2
Y
X1
Low X2
High X2
Y
X1
Low X2
High X2
Y
X1
Low X2
High X2
INTERACTION
NO INTERACTION
DuPont Quality Management and Technology SOE/ECHIP - 4.13© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
INTERACTION GEOMETRYX1 Effect at High and Low X2
X1
X2
X1*X2 Interaction Effect
= [(YD - YC) - (YB - YA)] / 2
(YA + YD) (YB + YC)=
2 2
YA
YB
YC
YD
DuPont Quality Management and Technology SOE/ECHIP - 4.14© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
INTERACTION GEOMETRYX2 Effect at High and Low X1
X1
X2
X1*X2 Interaction Effect
= [(YD - YB) - (YC - YA)] / 2
(YA + YD) (YB + YC)=
2 2
YA
YB
YC
YD
DuPont Quality Management and Technology SOE/ECHIP - 4.15© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: X1*X2 INTERACTION EFFECT
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
48
37
74
61 36
72 + 74 + 62 + 61 48 + 37 + 48 + 364 4
= 25
DuPont Quality Management and Technology SOE/ECHIP - 4.16© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: X1*X3 INTERACTION EFFECT
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
37 + 72 + 48 + 61 48 + 62 + 74 + 364 4
= -0.5
DuPont Quality Management and Technology SOE/ECHIP - 4.17© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: X2*X3 INTERACTION EFFECT
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
48 + 72 + 61 + 36 62 + 37 + 48 + 744 4
= -1
DuPont Quality Management and Technology SOE/ECHIP - 4.18© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
SAMPLE CALCULATION: X1*X2*X3 INTERACTION EFFECT
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
72 + 62 + 48 + 36 48 + 37 + 74 + 614 4
= -0.5
DuPont Quality Management and Technology SOE/ECHIP - 4.19© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
HALF OF THREE-FACTOR INTERACTION SHOWS BALANCE IN ALL FACTOR PAIRS
X2X1
X3
X3
X2
X1
DuPont Quality Management and Technology SOE/ECHIP - 4.20© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
USES OF HIGHER-ORDERINTERACTION GEOMETRY
Blocking– Basis for Splitting Experiment into
Smaller Blocks– Factors Are Balanced Within Blocks
Screening– Cut Experiment in Half by Using Only
One of Blocks– Factors Are Balanced
DuPont Quality Management and Technology SOE/ECHIP - 4.21© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
FACTORIAL DESIGNS: Summary
The “Cube” Approach Each Dimension Is a Factor Coding: Low = “-” and High = “+” Effects Are Comparisons of “Planes” Hidden Replication Efficiency: All Data Used to Calculate
Each Effect High-Order Interactions
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 5.1Revised 5/09/2000
Section 5
FACTORIAL EXAMPLE:Design
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SOE/MTB - 5.2Revised 5/09/2000
Follow the example to design a 2-level full-factorial experiment forthe workshop 1 problem.
Factors RangeCATALYST CONCENTRATION 1.0 to 1.8 REACTOR TEMPERATURE 130 to 190AMOUNT OF ADDITIVE 1 to 5
ResponsesCOLOR MODULUS
Design = Full-Factorial with center pointsModel = Linear + 2 factor interaction terms
FACTORIAL DESIGN EXAMPLEDesign Phase
Link to Workshop 1 Problem
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SOE/MTB - 5.3Revised 5/09/2000
X3
X2
X1
THREE-FACTOR CUBE PLOT
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SOE/MTB - 5.4Revised 5/09/2000
TEST FOR CURVATURE
+0-
YDifference IsCurvature
..............
X
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 5.5Revised 5/09/2000
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SOE/MTB - 5.6Revised 5/09/2000
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SOE/MTB - 5.7Revised 5/09/2000
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 5.8Revised 5/09/2000
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 5.9Revised 5/09/2000
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 5.10Revised 5/09/2000
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SOE/MTB - 5.11Revised 5/09/2000
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SOE/MTB - 5.12Revised 5/09/2000
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SOE/MTB - 5.13Revised 5/09/2000
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SOE/MTB - 5.14Revised 5/09/2000
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SOE/MTB - 5.15Revised 5/09/2000
DuPont Quality Management and Technology SOE/MTB - 6.1© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
Section 6
ANALYSIS OF TWO-LEVEL FACTORIAL DESIGNS
DuPont Quality Management and Technology SOE/MTB - 6.2© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SECTION 6 OVERVIEW
How do we know the effects are real ?
Computer analysis of the effects
How well have we explained the behaviorof Y?
DuPont Quality Management and Technology SOE/MTB - 6.3© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
HOW DO WE KNOW THE EFFECTS ARE “REAL” ?If each corner is an average of replicated runs, we can
study the repeatability of the effects.
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
Rep. 1
Rep. 262
48 72
37
48 74
61 36
DuPont Quality Management and Technology SOE/MTB - 6.4© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
EXAMINE THE REPLICATES FOR REPEATABILITY (Case 1)
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
4650
7371
4947
7573
3834
6260
6163
3638
DuPont Quality Management and Technology SOE/MTB - 6.5© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
EXAMINE THE REPLICATES FOR REPEATABILITY (Case 2)
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
5292
3363
44104
5418
2498
19105
5717
6630
DuPont Quality Management and Technology SOE/MTB - 6.6© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
THE IMPACT OF EXPERIMENTAL ERROR
Effect
AverageRelationship
Low (-) High (+)Factor
Response Experimental Error
DuPont Quality Management and Technology SOE/MTB - 6.7© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
HOW “REAL” ARE THE EFFECTS ?
Must estimate the size of the experimental error.
STANDARD DEVIATION: A measure of variability
(Y1- Y)2 + (Y2 - Y)2 + . . . + (Yn - Y)2
n - 1Std. Dev. = s =
DuPont Quality Management and Technology SOE/MTB - 6.8© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
EXAMINE THE REPLICATES FOR REPEATABILITY (Case 1)
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
4650
7371
4947
7573
3834
6260
6163
3638
s=2.8
s=1.4
s=2.8s=1.4
s=1.4
s=1.4
s=1.4s=1.4
DuPont Quality Management and Technology SOE/MTB - 6.9© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
POOLED STANDARD DEVIATION
Based on a weighted average of individual squared standard deviations
Assumes homogeneous error --- size ofexperimental error is uniform throughoutdesign region
A more reliable estimate of the overall standard deviation
(n1 - 1) s12 + (n2 - 1) s2
2 + . . . + (nk - 1) sk2
(n1 - 1) + (n2 - 1) + . . . + (nk - 1)spooled =
DuPont Quality Management and Technology SOE/MTB - 6.10© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
COMPUTE THE POOLED STANDARD DEVIATIONFOR THE 23 EXAMPLE: (Case 1)
spooled =
3.5 = 1.87
1*2 + 1*2 + 1*8 + 1*2 + 1*8 + 1*2 + 1*2 + 1*21 + 1 + 1 + 1 + 1 + 1 + 1 + 1
= 28
8
=
DuPont Quality Management and Technology SOE/MTB - 6.11© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
INTERPRETING THE STANDARD DEVIATION
-4 -3 -2 -1 0 1 2 3 4 SD units68%95%
99.7%
DuPont Quality Management and Technology SOE/MTB - 6.12© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
DISTRIBUTION OF AVERAGES
The distribution of the means of samples of size n is:–More nearly normal
–Narrower σY = σY n
DuPont Quality Management and Technology SOE/MTB - 6.13© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
THE STANDARD ERROR OF A FACTOR EFFECT
Represents the PRECISION in an estimated factor effect
Can be used to “test” if a factor effect is “statistically significant”
STD. ERROR = S-Pooled * SQRT(1/n1 + 1/n2)where
n1 = number of observations forming YLOW
n2 = number of observations forming YHIGH
DuPont Quality Management and Technology SOE/MTB - 6.14© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
EXAMPLE CALCULATION OF STANDARD ERROR
Consider the C*T interaction effect The estimated effect = 25 Standard error = 1.87 * SQRT(1/8 + 1/8) = 0.9
(1.87 is the pooled standard deviation calculatedearlier with 8 degrees of freedom)
Tabled t-value for 8 df, 95% confidence level
= 2.31 (t-table on next page)
DuPont Quality Management and Technology SOE/MTB - 6.15© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
t-DISTRIBUTION VALUES FOR TWO-SIDED CONFIDENCE INTERVALS
99%
63.709.925.844.604.03
3.713.503.363.253.17
3.113.053.012.982.95
2.922.902.882.862.85
2.832.822.812.802.79
2.752.702.662.622.58
95%
12.704.303.182.782.57
2.452.362.312.262.23
2.202.182.162.142.13
2.122.112.102.092.09
2.082.072.072.062.06
2.042.022.001.981.96
90%
6.312.922.352.132.01
1.941.891.861.831.81
1.801.781.771.761.75
1.751.741.731.731.72
1.721.721.711.711.71
1.701.681.671.661.64
DF
12345
6789
10
1112131415
1617181920
2122232425
304060
120∞
DuPont Quality Management and Technology SOE/MTB - 6.16© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SIGNIFICANCE OF AN EFFECTBY SIGNIFICANCE TEST
Consider the C*T interaction effect Observed t-ratio = (estimated effect)/(std error)
= 25.0 / 0.9 = 28 Compare observed t-ratio to tabled t-value:
Here |t-ratio| > tabled t-value (28 > 2.31)So the C*T interaction is statistically significantat the 0.95 (or 95%) confidence level
Note that the C*T interaction is also significant at the0.99 confidence level (28 > 3.36).
How high can we take the confidence level and still besignificant? This is related to the P-value.
DuPont Quality Management and Technology SOE/MTB - 6.17© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
P-VALUE
The P-value of an effect is the chance of having observed an effect that large purely due to random experimental error
The larger the effect, the smaller the P-value P-value is commonly displayed by most statistical
packages, usually as a decimal (i.e. between 0 and 1) P-value = 1 - (maximum confidence level at which
the effect is significant) EXAMPLE: P-value of an effect = 0.02
So the effect is significant at 98% confidence level For the C*T interaction effect, the P-value is 0.0000 !
DuPont Quality Management and Technology SOE/MTB - 6.18© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SIGNIFICANCE OF AN EFFECTBY CONFIDENCE INTERVAL
Consider the C*T interaction effect Confidence interval is:
Estimated effect +/- (t-value * standard error) For this example 95% confidence interval
is25 +/- (2.31 * 0.9) = 25 +/- 2.1 or 22.9 to 27.1
Whenever ZERO is not included in the confidence interval, the effect is significant at the specified confidence level
Graphical display of uncertainty
DuPont Quality Management and Technology SOE/MTB - 6.19© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SECTION 6 OVERVIEW
How do we know the effects are “real” ?– Estimate the experimental error (s-pooled)
and the standard error of the effect– Compare the estimated effect with its standard
error– t-test or– confidence interval
Computer analysis of the effects How well have we explained the
behavior of Y ?
DuPont Quality Management and Technology SOE/MTB - 6.20© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
COMPUTER ANALYSIS OF THEESTIMATED EFFECTS
Two common approaches: EFFECTS ANALYSIS
– Displays each effect and t-test or confidence interval
– Follows approach just covered earlier REGRESSION ANALYSIS
– Expresses each effect as a slope of a line or coefficient of a model term
– Statistically equivalent to an effects analysis
DuPont Quality Management and Technology SOE/MTB - 6.21© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
STANDARD FACTOR CODINGRegression coefficient (slope) = 1/2 effect
Effect
AverageRelationshipSlope = ∆ / 2
Low (-1) 0 High (+1)
Factor
Response
DuPont Quality Management and Technology SOE/MTB - 6.22© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
MEANING OF COEFFICIENTS:Temperature Effect
190180170160150140130
75
65
55
45
35
CO
LO
R
Temperature
130 140 150 160 170 180 190
35
45
5
65
75Slope of this line =Regression coefficient = 5.75(using centered/scaled factor settings)
For this example:Effect = 11.5 = expected
increase in COLOR fromTemperature=130 to 190(-1 to +1)
Coefficient = 5.75 = expectedincrease in COLOR fromTemperature=160 (middle)to 190 (0 to +1). Usesorthogonally scaled(-1 to +1) factor settings.
(-1) (+1)(0)
DuPont Quality Management and Technology SOE/MTB - 6.23© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Minitab Effects Table for COLOR Response
Fractional Factorial Fit: COLOR versus Catalyst, Temperature, Additive
Estimated Effects and Coefficients for COLOR (coded units)
Term Effect Coef SE Coef T P
Constant 54.7500 0.4488 122.00 0.000
Catalyst 0.0000 0.0000 0.4488 0.00 1.000
Temperat 11.5000 5.7500 0.4488 12.81 0.000
Additive 0.0000 0.0000 0.4488 0.00 1.000
Catalyst*Temperat 25.0000 12.5000 0.4488 27.85 0.000
Catalyst*Additive -0.5000 -0.2500 0.4488 -0.56 0.591
Temperat*Additive -1.0000 -0.5000 0.4488 -1.11 0.294
DuPont Quality Management and Technology SOE/MTB - 6.24© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
ITEMS IN THE MINITABEFFECTS & COEFFICIENTS TABLE
Term: Name for model term.
Effect: Expected change in the response over the entire range of the term (high “plane” - low “plane” averages).
Coef: Regression coefficients for the model terms. Expected change in the response per unit change in the term. Note carefully the centering and scaling used for these coefficients for interpretation.
SE Coef: Standard error of the coefficient -- uncertainty around the coefficients due to experimental error.
T: The ratio of the coefficients divided by their standard errors.
P: Probability of observing a coefficient of that magnitude when the true coefficient is zero. Low values (<.05) imply significance.
– If significant, we are pretty sure that at least the sign (direction) of the relationship is correct
DuPont Quality Management and Technology SOE/MTB - 6.25© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 Factor Example (Case 1)Graphical Representation of Factor Effects
Catalyst Temperature Additive
1.0 1.8 130 190 1 5
50.0
52.5
55.0
57.5
60.0
CO
LOR
Main Effects Plot (data means) for COLOR
DuPont Quality Management and Technology SOE/MTB - 6.26© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 Factor Example (Case 1)Graphical Representation of the Interactions
1 1.8 130 190 1 5
40
55
70
40
55
70
40
55
70Catalyst
Temperature
Additive
1
1.8
130
190
1
5
Interaction Plot (data means) for COLOR
DuPont Quality Management and Technology SOE/MTB - 6.27© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Minitab Cube Plot for COLOR Response
36
37
74
72
61
62
48
48
1.0 1.8
Catalyst
Temperature
Additive
130
190
1
5
Cube Plot (data means) for COLOR
DuPont Quality Management and Technology SOE/MTB - 6.28© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Pareto Plot of Effects on COLOR
DuPont Quality Management and Technology SOE/MTB - 6.29© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Normal Probability Plot for Effects on COLOR
0 10 20
-1
0
1
Standardized Effect
Nor
mal
Sco
re
AB
B
Normal Probability Plot of the Standardized Effects(response is COLOR, Alpha = .10)
A: CatalystB: TemperatC: Additive
DuPont Quality Management and Technology SOE/MTB - 6.30© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SECTION 6 OVERVIEW
How do we know the effects are “real” ?
Computer analysis of the effects– Effects analysis– Regression analysis
How well have we explained the behaviorof Y ?
DuPont Quality Management and Technology SOE/MTB - 6.31© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
TWO ESTIMATES OF EXPERIMENTAL ERROR
Pure Error Estimate (PE)– Based on variability among replicate runs under
fixed X-settings– Same as the pooled standard deviation
Lack of Fit Estimate (LOF)– Based on how well the estimated effects explain
the variation in the observed data.
When both the PE and LOF estimates are available, they can be used to test if additional effects shouldbe accounted for.
DuPont Quality Management and Technology SOE/MTB - 6.32© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
WHEN ARE P.E. AND L.O.F. ESTIMATES AVAILABLE ?
Not Available Available
NotAvailable
Available
Lack of Fit
Pure
Err
or
df: 2, 0, 0 df: 2, 0, 1
df: 2, 2, 0 df: 2, 3, 1
Key:df: #, #, #
Fit
PureError
Lack ofFit
DuPont Quality Management and Technology SOE/MTB - 6.33© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
THE ANALYSIS OF VARIANCE
Statistical technique for evaluating PURE ERROR and LACK-OF-FITexperimental error
Characterizes how well the estimatedeffects explain the data
DuPont Quality Management and Technology SOE/MTB - 6.34© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
ONE FACTOR EXAMPLEExperiment Worksheet
Row FACTOR RESPONSE1 -1 252 -1 353 0 604 0 805 1 456 1 55
DuPont Quality Management and Technology SOE/MTB - 6.35© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
ONE FACTOR EXAMPLEData Graph with Fitted Line
10-1
80
70
60
50
40
30
Factor X
Resp
onse
Y = 50 + 10XY
DuPont Quality Management and Technology SOE/MTB - 6.36© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
ONE FACTOR EXAMPLECoefficients and Analysis of Variance Results
The regression equation isRESPONSE = 50.0 + 10.0 FACTOR
Predictor Coef SE Coef T PConstant 50.000 7.906 6.32 0.003FACTOR 10.000 9.682 1.03 0.360
S = 19.36 . . .
Analysis of Variance
Source DF SS MS F PRegression 1 400.0 400.0 1.07 0.360Residual Error 4 1500.0 375.0Lack of Fit 1 1200.0 1200.0 12.00 0.041Pure Error 3 300.0 100.0
Total 5 1900.0
DuPont Quality Management and Technology SOE/MTB - 6.37© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
FURTHER EXPLANATION ON SELECTEDCOEFFICIENT TABLE ITEMS
Regression F-test: Tests whether any of the model terms explain the behavior of Y (overall test). If the model has some significant terms, the p-value should be small (<.05).
Lack of Fit F-test: Tests whether model can be improved with existing data. Might be significant due to missing terms, outliers, a need for a transformation, lack of measurement precision, etc. An appropriate model will have a non-significant lack of fit F-test (high p-value).
Square Root of MS Pure Error: Pooled standard deviation from replicates. Estimates SD of experimental error at any fixed set of conditions.
DuPont Quality Management and Technology SOE/MTB - 6.38© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
REGRESSION AND LACK OF FIT SIGNIFICANCEExample 1
Regression p-value = .0000Lack of fit p-value = .5514
FACTOR.X252015
14
12
10
8
6
4
2
0
FACTOR.X
RESP
.Y1
DuPont Quality Management and Technology SOE/MTB - 6.39© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
REGRESSION AND LACK OF FIT SIGNIFICANCEExample 2
Regression p-value = .8833Lack of fit p-value = .9641
252015
5.5
5.0
4.5
FACTOR.X
RESP
.Y2
DuPont Quality Management and Technology SOE/MTB - 6.40© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
REGRESSION AND LACK OF FIT SIGNIFICANCEExample 3
Regression p-value = .8184Lack of fit p-value = .0000
252015
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
FACTOR.X
RESP
.Y3
DuPont Quality Management and Technology SOE/MTB - 6.41© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
REGRESSION AND LACK OF FIT SIGNIFICANCEExample 4
Regression p-value = .0001Lack of fit p-value = .0003
252015
12.6
11.6
10.6
9.6
8.6
7.6
6.6
5.6
4.6
3.6
FACTOR.X
RESP
.Y4
DuPont Quality Management and Technology SOE/MTB - 6.42© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
BACK TO OUR 3 FACTOR EXAMPLE . . .Case 1
X3 (Additive)
X2 (Temperature)
X1 (Catalyst)
62
48 72
37
48 74
61 36
4650
7371
4947
7573
3834
6260
6163
3638
DuPont Quality Management and Technology SOE/MTB - 6.43© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Coefficients Table for COLOR Response
Fractional Factorial Fit: COLOR versus Catalyst, Temperature, Additive
Estimated Effects and Coefficients for COLOR (coded units)
Term Effect Coef SE Coef T P
Constant 54.7500 0.4488 122.00 0.000
Catalyst 0.0000 0.0000 0.4488 0.00 1.000
Temperat 11.5000 5.7500 0.4488 12.81 0.000
Additive 0.0000 0.0000 0.4488 0.00 1.000
Catalyst*Temperat 25.0000 12.5000 0.4488 27.85 0.000
Catalyst*Additive -0.5000 -0.2500 0.4488 -0.56 0.591
Temperat*Additive -1.0000 -0.5000 0.4488 -1.11 0.294
DuPont Quality Management and Technology SOE/MTB - 6.44© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
3 FACTOR EXAMPLE (Case 1)Analysis of Variance for COLOR Response
Analysis of Variance for COLOR (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 529.00 529.00 176.333 54.72 0.000
2-Way Interactions 3 2505.00 2505.00 835.000 259.14 0.000
Residual Error 9 29.00 29.00 3.222
Lack of Fit 1 1.00 1.00 1.000 0.29 0.608
Pure Error 8 28.00 28.00 3.500
Total 15 3063.00
DuPont Quality Management and Technology SOE/MTB - 6.45© 2000 E. I. du Pont de Nemours and Company Revised 1/29/2001
SECTION 6 OVERVIEW
How do we know the effects are “real” ?
Computer analysis of the effects
How well have we explained thebehavior of Y ?
– Analysis of variance results–Regression F-test–Lack of fit test
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.1Revised 1/29/2001
Section 7
FACTORIAL EXAMPLE:Analysis
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.2Revised 1/29/2001
Recall the workshop 1 problem (summarized below) for which we generated a design earlier in ECHIP. We already have response data entered so we’renow ready to analyze the data.
Factors RangeCATALYST CONCENTRATION REACTOR TEMPERATURE AMOUNT OF ADDITIVE
Design = Full-Factorial with center pointsModel = Linear + two-factor interaction terms
1.0 to 1.8130 to 190
1 to 5
FACTORIAL DESIGN EXAMPLEAnalysis Phase
ResponsesCOLORMODULUS
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.3Revised 1/29/2001
Additive
Temperature
Catalyst
THREE-FACTOR CUBE PLOT OFEXAMPLE WORKSHOP 1 DATA
6059
5857
4849
5051
3435
3638
7374
7271
38 4142 43
45
25
54
34
125
105
134
114
80
KEY: Numbers inside circles = COLOR valuesNumbers outside circles = MODULUS values
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.4Revised 1/29/2001
Workshop 1 RevisitedTeam
COLOR -- Coefficients (circle if significant)Cata Temp Addi C*T C*A T*A
Signif.of LOFtest
ReplicateStandardDeviation
123456789
10
Comments
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.5Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.6Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.7Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.8Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.9Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.10Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.11Revised 1/29/2001
Fractional Factorial Fit: COLOR versus CATALYST, TEMPERATURE, ADDITIVEEstimated Effects and Coefficients for COLOR (coded units)
Term Effect Coef SE Coef T PConstant 51.450 1.535 33.52 0.000CATALYST 0.125 0.063 1.716 0.04 0.972TEMPERAT 13.875 6.938 1.716 4.04 0.002ADDITIVE 0.125 0.062 1.716 0.04 0.972CATALYST*TEMPERAT 22.875 11.438 1.716 6.66 0.000 CATALYST*ADDITIVE 0.125 0.063 1.716 0.04 0.972TEMPERAT*ADDITIVE -0.125 -0.063 1.716 -0.04 0.972CATALYST*TEMPERAT*ADDITIVE -2.125 -1.063 1.716 -0.62 0.547
Analysis of Variance for COLOR (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 3 770.19 770.19 256.729 5.45 0.0132-Way Interactions 3 2093.19 2093.19 697.729 14.81 0.0003-Way Interactions 1 18.06 18.06 18.063 0.38 0.547Residual Error 12 565.51 565.51 47.126
Curvature 1 546.01 546.01 546.012 308.01 0.000Pure Error 11 19.50 19.50 1.773
Total 19 3446.95
Unusual Observations for COLOR
Obs COLOR Fit SE Fit Residual St Resid4 38.0000 51.4500 1.5350 -13.4500 -2.01R
R denotes an observation with a large standardized residual . . .
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.12Revised 1/29/2001
Fractional Factorial Fit: MODULUS versus CATALYST, TEMPERATURE, ADDITIVE
Estimated Effects and Coefficients for MODULUS (coded units)
Term Effect Coef SE Coef T PConstant 79.60 0.05774 1378.71 0.000CATALYST 80.00 40.00 0.06455 619.68 0.000TEMPERAT 9.00 4.50 0.06455 69.71 0.000ADDITIVE -20.00 -10.00 0.06455 -154.92 0.000CATALYST*TEMPERAT -0.00 -0.00 0.06455 -0.00 1.000CATALYST*ADDITIVE 0.00 0.00 0.06455 0.00 1.000TEMPERAT*ADDITIVE 0.00 0.00 0.06455 0.00 1.000CATALYST*TEMPERAT*ADDITIVE 0.00 0.00 0.06455 0.00 1.000
Analysis of Variance for MODULUS (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 3 27524.0 27524.0 9174.67 1E+05 0.0002-Way Interactions 3 0.0 0.0 0.00 * *3-Way Interactions 1 0.0 0.0 0.00 * *Residual Error 12 0.8 0.8 0.07
Curvature 1 0.8 0.8 0.80 Pure Error 11 0.0 0.0 0.00
Total 19 27524.8. . .
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.13Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.14Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.15Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.16Revised 1/29/2001
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.17Revised 1/29/2001
1.0 1.8
190190130130
70
60
50
40
TEMPERATURE
CATALYST
Mea
nInteraction Plot (data means) for COLOR
Centerpoint
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.18Revised 1/29/2001
ADDITIVETEMPERATURECATALYST
511901301.81 .0
120
100
80
60
40
MO
DU
LUS
Main Effects Plot (data means) for MODULUSCenterpoint
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.19Revised 1/29/2001
ADDITIVETEMPERATURECATALYST
511901301.81 .0
60
55
50
45
40
CO
LOR
Main Effects Plot (data means) for COLORCenterpoint
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 7.20Revised 1/29/2001
DuPont Quality Management and Technology SOE/ECHIP - 8.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Section 8
GOOD EXPERIMENTALPRACTICE
DuPont Quality Management and Technology SOE/ECHIP - 8.2© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
GOOD EXPERIMENTAL PRACTICE
1. Assess the environment2. Consider the factors3. Consider the responses4. Choose an appropriate design5. Consider strategies for bias error6. Create the experimental plan7. Review the plan for operability8. Avoid blunders9. Plan for the analysis
10. Report the recommendations
DuPont Quality Management and Technology SOE/ECHIP - 8.3© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
1. ASSESS THE ENVIRONMENT
Gather basic information Determine current state of understanding Define experimental objectives Define physical environment and constraints Consider experimental error
DuPont Quality Management and Technology SOE/ECHIP - 8.4© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
2. CONSIDER THE FACTORS
How many and which factors to vary? Which factors to keep fixed, and at what
level? Are factors continuous or discrete? How many levels of each factor? How bold in choice of levels?
DuPont Quality Management and Technology SOE/ECHIP - 8.5© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
3. CONSIDER THE RESPONSES
Consider all responses of potential interest Are responses continuous or discrete? What is effect size of interest? Is measurement error absolute or relative? Anticipate measurement issues
DuPont Quality Management and Technology SOE/ECHIP - 8.6© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
4. CHOOSE AN APPROPRIATE DESIGN
Identify underlying model Choose appropriate design class Consider desired sensitivity (resolution) in
choosing size of design Consider extra runs Practical constraints?
DuPont Quality Management and Technology SOE/ECHIP - 8.7© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
NUMBER OF RUNS VS. SENSITIVITY
SOE Rule of Thumb for balanced 2-level factorial designs:
n = 7 or 8∆ / σ( )2
∆ = smallest size effect worth detectingσ = standard deviation of experimental error
∆ / σ is “signal-to-noise ratio”
∆ / σ 0.5 1.0 1.5 2.0
n 196-256 49-64 22-28 12-16
DuPont Quality Management and Technology SOE/ECHIP - 8.8© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
5. STRATEGIES FOR BIAS ERROR
Consider blocking to protect against bias due to identifiable variables
– e.g. time, material batch, operator Use randomization to protect against bias
due to unidentified variables Consider possible constraints on
randomization, do restricted randomization if necessary
Assess randomization, adjust if necessary
DuPont Quality Management and Technology SOE/ECHIP - 8.9© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
6. CREATE THE EXPERIMENTAL PLAN
Create experimental worksheet, in randomized run order
Express factor levels in actual physical units
Include blank columns for responses Write detailed protocol with explicit
instructions for every step
DuPont Quality Management and Technology SOE/ECHIP - 8.10© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
7. REVIEW PLAN FOR OPERABILITY
Include all relevant parties in review– Planner– Executor– Lab analyst– Local experts– Statistician
Examine all runs for operability, adjust if necessary
Consider constraints on experiment, feasibility of schedule
DuPont Quality Management and Technology SOE/ECHIP - 8.11© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
8. AVOID BLUNDERS
Anticipate problems in advance Emphasize importance of adhering to
experimental protocol Avoid shortcuts Understand importance of each run Avoid stopping short Record any deviations from plan
DuPont Quality Management and Technology SOE/ECHIP - 8.12© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
9. PLAN FOR THE ANALYSIS
Consider analysis issues in advance– Models– Software– Method of analysis– Organization of data
Plot the data
DuPont Quality Management and Technology SOE/ECHIP - 8.13© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
10. REPORT RECOMMENDATIONS
Relate recommendations to the objective Recognize value of negligible effects Avoid extrapolation Report raw data for credibility Provide direction for future action
DuPont Quality Management and Technology SOE/MTB - 9.1© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
Section 9
SCREENING DESIGNS
DuPont Quality Management and Technology SOE/MTB - 9.2© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
EVOLUTION OF THE ENVIRONMENT:Early Stage
INTERACTION RESPONSEDESIGNS SURFACE
DESIGNS
Evolution of the Experimental Environment
NUMBER OF 6 or more 3 - 8 2 - 6FACTORS
OBJECTIVE Identify key factors Understand factor Prediction modelinteractions Optimization
COMMON Plackett-Burman Full Factorial Box-BehnkenDESIGNS Fractional Factorial Fractional Factorial Central Composite
(resolution 3 or 4) (resolution 5) Face Center Cube
SCREENINGDESIGNS
DuPont Quality Management and Technology SOE/MTB - 9.3© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
CHARACTERISTICS OF SCREENING DESIGNS
Number of runs n only a few more than number of factors k
Factors considered at two levels each Use a fraction of the full 2k factorial design Designs balanced (orthogonal) Main factor effects clear of each other Interactions generally not estimable Two-way interactions may be fully or
partially confounded with main effects
DuPont Quality Management and Technology SOE/MTB - 9.4© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
CLASSES OF SCREENING DESIGNS
Plackett-Burman (P-B) designs
Fractional Factorial (FF) designs
DuPont Quality Management and Technology SOE/MTB - 9.5© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
PLACKETT-BURMAN DESIGNS
Available in sizes n which are multiples of 4 Can handle up to n-1 factors, although
recommended maximum is n-5 Tables available for commonly used sizes of
n=12, 20, 24, 28 For n equal to power of 2 (e.g. 8, 16, 32,...),
same as Fractional Factorial designs Two-way interactions partially confounded
with each main effect (for n not a power of 2)
DuPont Quality Management and Technology SOE/MTB - 9.6© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
12 RUN PLACKETT-BURMAN DESIGNTrial X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
1 + + - + + + - - - + -2 + - + + + - - - + - +3 - + + + - - - + - + +4 + + + - - - + - + + -5 + + - - - + - + + - +6 + - - - + - + + - + +7 - - - + - + + - + + +8 - - + - + + - + + + -9 - + - + + - + + + - -10 + - + + - + + + - - -11 - + + - + + + - - - +12 - - - - - - - - - - -
last row all minuses
each rowa cyclicpermutationof previousrow
DuPont Quality Management and Technology SOE/MTB - 9.7© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
BALANCE OF 12 RUN P-B DESIGN
Illustrate using factors X1 and X2Rows rearranged according to level of X1
Trial X1 X21 + +2 + -4 + +5 + +6 + -
10 + -
3 - +7 - -8 - -9 - +
11 - +12 - -
equal numberof + and -
equal numberof + and -
SAME BALANCE TRUEFOR ANY PAIR OF FACTORS(COLUMNS ARE “ORTHOGONAL”)
DuPont Quality Management and Technology SOE/MTB - 9.8© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
PARTIAL CONFOUNDING IN 12 RUN P-B
Illustration showing partial confounding between X1 and X2*X3 interaction.Rows rearranged according to level of X1.
Trial X1 X2 X3 X2*X31 + + - -2 + - + -4 + + + +5 + + - -6 + - - +10 + - + -
3 - + + +7 - - - +8 - - + -9 - + - -
11 - + + +12 - - - +
2 +’s4 -’s
4 +’s2 -’s
not completely balanced
If there is an X2*X3 interaction effect, it will slightly bias the estimate of the X1 effect(as well as all other main effects except X2 and X3).
DuPont Quality Management and Technology SOE/MTB - 9.9© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
USING PLACKETT-BURMAN DESIGNS
If using tabled design, assign factors to any columns of the design
If blocking, use additional column(s) to define blocks
Save at least 4 columns for estimating experimental error
Assess sensitivity of design --- if inadequate:– Use larger design– Add reflected design
DuPont Quality Management and Technology SOE/MTB - 9.10© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
FRACTIONAL FACTORIAL DESIGNS
Available design sizes in powers of 2 For k factors, 2k-p FF design is a 1/2p fraction
of a full 2k factorial design 16 and 32 run designs most useful for
screening Confounding between any two effects is
either total or absent Degree of confounding determined by
Resolution of design
DuPont Quality Management and Technology SOE/MTB - 9.11© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
CONSTRUCTING FF DESIGNS
TO CONSTRUCT 2k-p DESIGN, WHERE k = NUMBER OF FACTORS:
1. Construct a full factorial design in k-p of the factors.
2. Define each of the p additional factors as equal to, or the negativeof, a judiciously selected interaction among some of the first k-pfactors. These are called the defining relations of the fractionalfactorial design.
For given values of k and p, tables are available of good choicesfor the defining relations which will result in the highest possibleresolution design.
DuPont Quality Management and Technology SOE/MTB - 9.12© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
EXAMPLE 1: CONSTRUCTING A 24-1 DESIGN
4 FACTORS --- FOR SIMPLICITY, DENOTE FACTORS BY 1, 2, 3, 41. Write down full factorial 23 design in factors 1, 2, 32. Define factor 4 equal to 123 interaction1 2 3 4=123 Defining Relation 4=123 induces- - - - additional relations:+ - - + 1=234 2=134 3=124- + - + 12=34 13=24 14=23+ + - - 1234=I (I =column of all +)- - + + NOTE THAT EFFECTS ARE+ - + - CONFOUNDED IN PAIRS:- + + - main effects with 3-way+ + + + 2-way with 2-way
4-way with meanTHIS IS A RESOLUTION 4 DESIGN
DuPont Quality Management and Technology SOE/MTB - 9.13© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
EXAMPLE 2: CONSTRUCTING A 25-2 DESIGN
5 FACTORS --- 1, 2, 3, 4, 51. Write down full factorial 23 design in factors 1, 2, 32. Defining relations: 4 =12 , 5=13 1 2 3 4=12 5=13 Defining Relations induce- - - + + additional relations:+ - - - - 1=24=35=12345 2=14=345=1235- + - - + 3=15=245=1234 4=12=235=1345+ + - + - 5=13=234=1245 23=45=134=125- - + + - 25=34=123=145 I=124=135=2345+ - + - + NOTE THAT EFFECTS ARE NOW - + + - - CONFOUNDED IN GROUPS OF 4+ + + + + Each main effect is now confounded
with a 2-way interactionTHIS IS A RESOLUTION 3 DESIGN
DuPont Quality Management and Technology SOE/MTB - 9.14© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
RESOLUTION OF A DESIGN
Let R denote resolution of design.Two effects of order a and b are unconfounded if a+b < R.However, if a+b ≥ R, then the effects may be confounded.
In particular, a main effect is unconfounded with any effects oforder less than R-1 but may be confounded with an effect of orderequal to R-1.
RESOLUTION PROPERTIES OF DESIGN
3 Main effects clear of each other but confounded with some2-way interactions
4 Main effects clear of each other and of 2-way interactionsbut 2-way interactions confounded with each other
5 or more Main effects and 2-way interactions all clear of each other
DuPont Quality Management and Technology SOE/MTB - 9.15© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
AVAILABLE DESIGN SIZES
# OF FRACTIONAL FACTORIAL PLACKETT-BURMAN FACTORS R=3 R=4 R=5 MIN ≥ 5 DF ERROR
5 8 16 16 8 12
6 8 16 32 8 12
7 8 16 64 8 16
8 16 16 64 12 16
9 16 32 128 12 16
10 16 32 128 12 16
11 16 32 128 12 20
12 16 32 256 16 20
Note: All designs listed assuming no replicates and no center points
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 10.1Revised 9/26/2000
Section 10
SCREENING EXAMPLE
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 10.2Revised 9/26/2000
Screening Design Example
This example is a 2-level screening experiment for a product called GLOOP.Factors Abbreviation SettingsTENSION CONTROL TENS manual, automatic
MACHINE MACH #1, #2THROUGHPUT TPUT 10 to
20 gal/min MIXING MIX single, double TEMPERATURE TEMP 200°to 250° F MOISTURE MOIST 20% to 80%Response Abbreviation Expected RangePRODUCT HARDNESS Hardness 10 to 200 (Gauge)
Design = Plackett-Burman Model = Main Effects onlySpecial Notes: In the past day-to-day differences with this process have been observed. Since we can only make 8 product items per day we would also like to block this design on day. Thus, this study really involves 7 factors including the blocking variable (DAY). A preliminary standard deviation estimate of 13 for hardness has been obtained. Detecting changes in hardness of at least 30 units is of interest.
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 10.3Revised 9/26/2000
What Design Size Is Needed?
Using our sample size formula we need at least:
The smallest Plackett-Burman design for n ≥ 10 is the 12 run design
The smallest Fractional-Factorial design for n ≥10 is the 16 run design
n 730/13
9.2 10
n 10
2
≥
= ⇒
≥
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
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SOE/MTB 10.15Revised 9/26/2000
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Fractional Factorial Fit: Hardness versus TENS, MACH, ...
Estimated Effects and Coefficients for Hardness (coded units)
Term Effect Coef SE Coef T PConstant 70.00 3.592 19.49 0.000TENS 1.67 0.83 3.592 0.23 0.828MACH 19.33 9.67 3.592 2.69 0.055TPUT 26.33 13.17 3.592 3.67 0.021MIX -0.67 -0.33 3.592 -0.09 0.931TEMP 61.67 30.83 3.592 8.58 0.001MOIST 7.00 3.50 3.592 0.97 0.385DAY -20.00 -10.00 3.592 -2.78 0.050
Analysis of Variance for Hardness (coded units)
Source DF Seq SS Adj SS Adj MS F PMain Effects 7 15966.7 15966.7 2281.0 14.73 0.010Residual Error 4 619.3 619.3 154.8Total 11 16586.0. . .
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 10.26Revised 9/26/2000
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Thought Questions
What would happen if you analyzed this dataignoring the day effect?
Can we get any information on interactionsamong the significant effects with this
data?
DuPont Quality Management and Technology SOE/MTB - 11.1© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Section 11
WORKSHOP 2:Glyxel Screening
DuPont Quality Management and Technology SOE/MTB - 11.2© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Use a screening design to determine which of the following twelve factors are the most important in their effects on the two observed responses.
RESPONSES Bend Resistance (BR) -- valid range approximately 50-80 psigoal = target to 68 (acceptable range is 65-71)
Area Shrinkage (AS) -- valid range approximately 0-3 %goal = minimize (a maximum of 1% is desirable)
FACTORS LOW LIMIT HIGH LIMIT UNITSThroughput (TP) 200 800 kg/hr
Additive A Concentration (AC) 4 8 %Additive A Impurity (AI) 0.7 3.2 % impurity
Catalyst Amount (CA) 0.1 0.3 %Reactor Pressure (RP) 100 150 psiDryer Temperature (DT) 120 150 °C
Extruder Temperature (ET) 180 190 °CQuench Water Temperature (QT) 10 15 °CQuench Water Flow Rate (QF) 10 20 l/minPress Temperature (PT) 140 160 °CStorage Temperature (ST) 10 25 °CBlock (BL) - see description 2 blocks needed week
Workshop 2 - Problem Description
DuPont Quality Management and Technology SOE/MTB - 11.3© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Workshop 2 - Background Information - I
Your R&D team has been assigned to develop a new product for a critical aerospace use. There is also a potential market in the military aeronautic and automotive industries.The product, called Glyxel, will be produced and sold in sheet form. Two critical property goals must be met.
Bend Resistance (BR) - (acceptable range 65 - 71 psi, target of 68) Gives a proper balance between end-use strength and customer processing needs.
Area Shrinkage (AS) - (1.0% maximum, lower is better) Low shrinkage is required to maintain dimensional stability through customers’ processing.
Below are brief variable descriptions and key learnings from early R&D work. First, to meet the expected market demand, property goals should be met with the highest possible throughput (TP). It is expected that at least 500 kg/hr will be needed for acceptable profitability.Preliminary R&D work has indicated two process variables likely to aid in meeting bending resistance and area shrinkage goals are reactor pressure (RP) and extruder temperature (ET).Additive A concentration (AC) is suspected to be important for obtaining low area shrinkage. Impurity levels of this additive (AI) are also believed to affect area shrinkage. Impurity levels are dictated by the lot number. From our supplier we receive an an accurate estimate of impurity level through a Certificate Of Analysis for each lot. However, since we have 20 lots to choose from that span the range of impurity levels listed, you may experiment with impurity level and treat it as a continuous variable.Two other variables, quench water temperature (QT) and quench water flow rate (QF) have been suggested as potentially affecting either bending resistance or area shrinkage.
DuPont Quality Management and Technology SOE/MTB - 11.4© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
It is expected that, in operation, quench water would be supplied from a well with an average yearly temperature range from 10 to 15 deg C. However, you have a temperature controlled water supply (+/- 1 deg) available for experimental use. Hopefully, your team can better understand how typical quench water temperature variation will affect the process. Another related variable that may enhance quenching is quench water flow rate (QF).
The catalyst has been thought to enhance molecular structure development. Since our catalyst is expensive, a lower level of CA is desirable if property goals can be satisfied.
Materials exiting the reactor are fed into a dryer to remove moisture prior to extrusion. Dryer temperature (DT) may influence final product properties as well.
Once the product is formed into sheets, it is stored for up to two weeks in a warehouse until shipped. High storage temperature (ST) may be related to product property deterioration.
After our customer receives our product in sheet form, it is fed into presses for final shaping. The press design temperature (PT) is 150 deg C. However, due to press-to-press differences and temperature control, the actual temperature can vary from 140 to 160 deg C. Your R&D team has acquired a similar press for the duration of the experimental program.
Lastly, due to the complicated nature of this process and time required to make process and recipe changes, it is only possible to experiment with about 12 run combinations in a week. Because a knowledgeable technician has suggested product properties vary from week to week, we would like to test this claim and, more importantly, remove this potential source of variation by blocking (BL) the experiment. Since it is not possible to complete our 12-factor screening design in one week, you will need to run it over two weeks or blocks.
Workshop 2 - Background Information - II
DuPont Quality Management and Technology SOE/MTB - 11.5© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Workshop 2 - Screening Design Assignment
Choose a Plackett-Burman screening design to identify the most important factors affecting BR and AS. The smallest Plackett-Burman design which could handle 12 factors is the 16 run design which is actually a fractional-factorial design. As this would only have 3 degrees of freedom to estimate experimental error and should be treated as a resolution 3 fractional-factorial design (which are more complex to deal with), the next design might be preferable. This will be a 20-run design (with zero replicates). Make sure you define factors and responses by the 2-letter abbreviations (factors =>TP, AC, AI, ... ,BL responses => BR, AS).
Preliminary standard deviation estimates for BR and AS are 1.5 psi and 0.1 %, respectively. Assuming we are interested in detecting effects (least important difference) of at least 3.0 psi for BR and 0.2 % for AS, will our proposed screening design will have adequate sensitivity?
Type BR and AS as the names of two empty columns in the worksheet. Generate the response data using the simulator: %GLYXEL Examine the results in detail. Be prepared to answer the following questions and report your teams
results (see team report spreadsheets). Which factors have significant effects on product properties? What was the experimental error from
your results? Perhaps the most important question is: What factors have you selected to be included in further process optimization work? You may identify up to 5 factors to carry forward to the next design stage. Fill in your team results on the appropriate row of the team report spreadsheets. Note, we will pick up here in the next workshop where we will follow up with a response surface design to optimize this process.
DuPont Quality Management and Technology SOE/MTB - 11.6© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Workshop 2: Glyxel Screening - Team Reports
Effects for BEND RESISTANCE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL
1
2
3
4
5
6
7
8
9
10
Exp. Error(Resid. SD)
DuPont Quality Management and Technology SOE/MTB - 11.7© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Effects for BEND RESISTANCE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL
11
12
13
14
15
16
17
18
19
20
Exp. Error(Resid. SD)
Workshop 2: Glyxel Screening - Team Reports
DuPont Quality Management and Technology SOE/MTB - 11.8© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Effects for AREA SHRINKAGE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL
1
2
3
4
5
6
7
8
9
10
Exp. Error(Resid. SD)
Workshop 2: Glyxel Screening - Team Reports
DuPont Quality Management and Technology SOE/MTB - 11.9© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Effects for AREA SHRINKAGE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL
11
12
13
14
15
16
17
18
19
20
Exp. Error(Resid. SD)
Workshop 2: Glyxel Screening - Team Reports
DuPont Quality Management and Technology SOE/MTB - 11.10© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Circle Factors To Be Studied in Workshop 3Team
1
2
3
4
5
6
7
8
9
10
COMMENTS
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
(note that a maximum of 5 factors may be circled)
Workshop 2: Glyxel Screening - Team Reports
DuPont Quality Management and Technology SOE/MTB - 11.11© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
Circle Factors To Be Studied in Workshop 3Team
11
12
13
14
15
16
17
18
19
20
COMMENTS
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
TP AC AI CA RP DT ET QT QF PT ST BL
(note that a maximum of 5 factors may be circled)
Workshop 2: Glyxel Screening - Team Reports
DuPont Quality Management and Technology SOE/MTB - 12.1© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
Section 12
RESPONSE SURFACE DESIGNS
DuPont Quality Management and Technology SOE/MTB - 12.2© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
SCREENING INTERACTIONDESIGNS DESIGNS
Evolution of the Experimental Environment
NUMBER OF 6 or more 3 - 8 2 - 6FACTORS
OBJECTIVE Identify key factors Understand factor Prediction modelinteractions Optimization
COMMON Plackett-Burman Full Factorial Box-BehnkenDESIGNS Fractional Factorial Fractional Factorial Central Composite
(resolution 3 or 4) (resolution 5) Face Center Cube
EVOLUTION OF THE ENVIRONMENT:Advanced Stage
RESPONSESURFACEDESIGNS
DuPont Quality Management and Technology SOE/MTB - 12.3© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
USES OF RESPONSE SURFACE MODELS
Quantitative Understanding
Prediction
Optimization
Conditions for Stability
Calibration
Process Control Adjustments
DuPont Quality Management and Technology SOE/MTB - 12.15© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
FACE-CENTERED CUBE DESIGN for 3 factors
X3
X2
X1
Block 1(First Half-Fraction)
Block 2(Second Half Fraction)
Block 3(Face Points)
Center Points
DuPont Quality Management and Technology SOE/MTB - 12.18© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
BOX-BEHNKEN DESIGNfor 3 factors
X3
X2
X1
Edge Centers
Center Point
DuPont Quality Management and Technology SOE/MTB - 12.22© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
SETTING UP A RESPONSESURFACE EXPERIMENT
Assess the Environment Consider the Factors Consider the Responses Choose an Appropriate Design Consider Strategies for Bias Error Rewrite the Experimental Schedule in
Physical Units Review the Experiment for Operability Avoid Blunders Plan for the Analysis Report the Recommendations
DuPont Quality Management and Technology SOE/MTB - 12.28© 2000 E. I. du Pont de Nemours and Company Revised 9/26/2000
RESPONSE SURFACE DESIGNS: Summary
Quadratic Polynomial Models
Danger of Extrapolation
Shape of Experimental Region
– Cubical: Face-Centered Cube
– Spherical: Central Composite, Box-Behnken
Space-Filling, Balanced, and Robust
Link to Model Diagnostics
Link to RS Example
Link to RS Workshop
DuPont Quality Management and Technology SOE/MTB - 13.2© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
UNDERLYING ASSUMPTIONS
Fitted model form is “approximately correct”– More important for Response Surface models
Deviations from model have no systematic component (bias error)
Experimental error (random error) is approximately normally distributed
SD of experimental error is homogeneous throughout experimental space
DuPont Quality Management and Technology SOE/MTB - 13.3© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
DIAGNOSTICS AND REMEDIES
ASSUMPTION DIAGNOSTIC TOOL POSSIBLE REMEDY
Correct model form Lack-of-fit test Model augmentation
Transformation
No bias error Residual plots Bias modeling
Outlier handling
Normal error Residual histogram Transformation& normal plot
Fundamental knowledge
Homogeneous error Residual plots Transformation
Fundamental knowledge Weighted regression
DuPont Quality Management and Technology SOE/MTB - 13.12© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
RESIDUAL PLOTS IN MINITAB
Minitab can calculate 3 kinds of residuals: Regular, Standardized, or Studentized [“Deleted”]. Standardized and “Deleted” residuals use a standard deviation scale.
Available plots of residuals include:HistogramNormal Probability Plot vs. Time vs. Fits (Predicted Values) vs. Variables
DuPont Quality Management and Technology SOE/MTB - 13.13© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Minitab: Transformations
• Transformations in Minitab are calculated using the Calculator… item in the Calcpull-down menu.
• Store results of transformations in separatecolumns so that original values are retained.
DuPont Quality Management and Technology SOE/MTB - 13.14© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
R-Squared & Adjusted R-Squared
Both estimate percent or proportion of observed variability explained by model
R-squared adjusted is generally a more honest appraisal as it adjusts for the number of terms in the model.
Minitab will not display R-Squared values for the analyses of 2-level designs but will display them for response surface designs.
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 14.1Revised 1/30/2001
Section 14
RESPONSE SURFACE EXAMPLE
Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB - 14.2Revised 1/30/2001
A new compound is being developed for a coating process. Threefactors are under consideration:
Factor RangeADDITIVE AMOUNT 0 to 70 gramsREACTION TIME 20 to 60 minutesREACTION TEMPERATURE 100 to 180 degrees C
The yield of the compound is measured. The compound is added to a fixed amount of formula and the coating process completed. Adhesion is then measured. Specifications on the responses are:
YIELD ≥ 91%ADHESION ≥ 45 grams
Find settings of the factors for which these conditions can be achieved. Resources are available for a maximum of 24 runs.
RESPONSE SURFACE EXAMPLEProblem Statement
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EXAMPLE DATA PLOTTED ON CUBES
Tem
pera
ture
Additive
Tem
pera
ture
Additive
YIELD ADHESION
68 3
82,87,87,82,85,85
37,41,40,40,42,42
40 37
50 4090 39
81 1065 48
51 40
68 2475 4480 38
92 41
75 31
77 44
75 31
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Response Surface Regression: YIELD versus Additive, Time, Temperature
The analysis was done using coded units.
Estimated Regression Coefficients for YIELD
Term Coef SE Coef T PConstant 84.55 0.6964 121.403 0.000Additive 4.90 0.6406 7.649 0.000Time 6.40 0.6406 9.991 0.000Temperat -0.80 0.6406 -1.249 0.240Additive*Additive -12.86 1.2216 -10.530 0.000Time*Time 1.64 1.2216 1.340 0.210Temperat*Temperat -8.36 1.2216 -6.847 0.000Additive*Time 0.75 0.7162 1.047 0.320Additive*Temperat 13.50 0.7162 18.849 0.000Time*Temperat -0.25 0.7162 -0.349 0.734
S = 2.026 R-Sq = 98.9% R-Sq(adj) = 97.9%
Analysis of Variance for YIELD
Source DF Seq SS Adj SS Adj MS F PRegression 9 3746.71 3746.71 416.302 101.45 0.000Linear 3 656.10 656.10 218.700 53.29 0.000Square 3 1627.61 1627.61 542.538 132.21 0.000Interaction 3 1463.00 1463.00 487.667 118.84 0.000
Residual Error 10 41.04 41.04 4.104 Lack-of-Fit 5 15.70 15.70 3.141 0.62 0.694Pure Error 5 25.33 25.33 5.067
Total 19 3787.75
...
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Response Surface Regression: ADHESION versus Additive, Time, TemperatureThe analysis was done using coded units.Estimated Regression Coefficients for ADHESION
Term Coef SE Coef T PConstant 40.24 0.5847 68.816 0.000Additive 8.80 0.5378 16.362 0.000Time 2.90 0.5378 5.392 0.000Temperat 5.90 0.5378 10.970 0.000Additive*Additive -6.09 1.0256 -5.939 0.000Time*Time -0.59 1.0256 -0.576 0.577Temperat*Temperat -2.59 1.0256 -2.526 0.030Additive*Time 0.75 0.6013 1.247 0.241Additive*Temperat -10.25 0.6013 -17.046 0.000Time*Temperat -0.50 0.6013 -0.831 0.425
S = 1.701 R-Sq = 98.8% R-Sq(adj) = 97.7%
Analysis of Variance for ADHESION
Source DF Seq SS Adj SS Adj MS F PRegression 9 2399.87 2399.87 266.653 92.18 0.000Linear 3 1206.60 1206.60 402.200 139.04 0.000Square 3 346.27 346.27 115.424 39.90 0.000Interaction 3 847.00 847.00 282.333 97.60 0.000
Residual Error 10 28.93 28.93 2.893 Lack-of-Fit 5 11.59 11.59 2.319 0.67 0.665Pure Error 5 17.33 17.33 3.467
Total 19 2428.80
Unusual Observations for ADHESIONObservation ADHESION Fit SE Fit Residual St Resid
3 37.000 40.236 0.585 -3.236 -2.03RR denotes an observation with a large standardized residual. ...
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Do any of the initial design points satisfy the specified performance criteria for YIELD and ADHESION ?
Which model terms are statistically significant ? What confidence level did you use to judge significance ?
How well does the model fit the data ? How did you assess this ?
At a given set of experimental conditions (X settings), what is the experimental error in YIELD and ADHESION ?
THOUGHT QUESTIONSPart 1
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Click on OK and then OK againin the next frame to get to:
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706050403020100
180
170
160
150
140
130
120
110
100
Additive
Tem
pera
ture
Hold values: Time: 60.0 Hold values: Time: 60.0 Hold values: Time: 60.0 Hold values: Time: 60.0
Overlaid contours for desired values of Yield, Adhesion
YIELD
ADHESION
91100
45100
Lower BoundUpper Bound
White area: feasible region
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Link to RS Workshop
DuPont Quality Management and Technology SOE/MTB - 15.1© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Section 15
WORKSHOP 3:Glyxel Response Surface
DuPont Quality Management and Technology SOE/MTB - 15.2© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
This workshop is a continuation of the Glyxel problem described in workshop 2, however, now we will use a response surface design to optimize the process. Here we will study in more detail the key factors identified through your screening design.
The original list of factors examined in workshop 2 are given below. From your previous assignment you reduced this list down to the critical few variables (up to 5) to investigate in this response surface design.
Factors (original list) Low Limit High Limit UnitsThroughput (TP) 200 800 kg/hr
Additive A Concentration (AC)4 8% Additive A
Impurity (AI) 0.7 3.2 % impurityCatalyst Amount (CA) 0.1 0.3 %
Reactor Pressure (RP)100 150 psiDryer Temperature (DT) 120 150 °C
Extruder Temperature (ET) 180 190 °C
Quench Water Temperature (QT) 1015 °C Quench
Water Flow Rate (QF) 10 20 l/minPress Temperature (PT) 140 160 °C
St T t (ST)
Workshop 3 - Problem Description
DuPont Quality Management and Technology SOE/MTB - 15.3© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 - Background Information
If needed, refer back to the original problem description (Section 11 - workshop 2 background information slides) to refresh your memory on the design variable descriptions.
Recall the two critical property goals.Bend Resistance (BR) - (acceptable range 65 - 71 psi, target of 68) Gives a proper
balance between end-use strength and customer processing needs.Area Shrinkage (AS) - (1.0% maximum, lower is better) Low shrinkage is required to
maintain dimensional stability through customers’ processing.
Additionally, recall that based on acceptable profitability and market demand expectations we need to satisfy property goals with the highest possible throughput. It was previously stated that we needed to achieve at least 500 kg/hr if possible with higher levels being more desirable.
Several of the factors originally described fall into the category of environmental variables. Such variables may include uncontrolled, noise, ambient, raw materials, or customer use variables. Although environmental variables are not typically controlled in operation, we may choose to explicitly control them within the context of a designed experiment to understand their potential impact on product properties.
Which factors explored in the Glyxel problem in workshop 2 are environmental variables? It is also possible that some factors retained for the optimization study in this workshop are environmental variables. What are they? The importance of Identifying environmental factor(s) and understanding their nature and how they might be treated during optimization will become clear in the assignment discussion and thought questions.
DuPont Quality Management and Technology SOE/MTB - 15.4© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 - RS Design Assignment (Part 1)
Select a response surface design. Choose either a CENTRAL COMPOSITE or BOX-BEHNKEN design. Design sizes will vary according to the number of factors studied, design type, and the number of replicates.
Verify that your intended design will be large enough to detect the size effects we are interested in. Recall that you want to detect a 3 psi change in BR and a 0.2% change in AS. You now have 2 standard deviation estimates for BR and AS, the preliminary estimates (given in workshop 2) and the estimates obtained from the screening design. Which estimate would you use in your design size or sensitivity calculations here and why?
Type BR and AS as column names for two unused columns in the worksheet. Generate the response data with the simulator by typing: %GLYXELRS
in the session window. This simulator will ask for settings for your fixed factors. Enter * for factors that are in the design.
Proceed to analyze.
Examine the results in detail to understand which effects are important and how well the model fits. Fill in your teams results on the appropriate row in the BR and AS regression model results spreadsheets and answer all thought questions (slide 7) prior to generating any contour plots.
DuPont Quality Management and Technology SOE/MTB - 15.5© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 - RS Design Assignment (Part 2)
Generate a few contour plots to get a feel for the behavior of the response surfaces. Keep in mind that knowing the key model effects can help point you to a more promising portion of the design space. Try to identify design regions where property goals can be satisfied.
Initially you may want to assume that any environmental factor(s) will vary across the full design range. Under this assumption you might begin by generating contour plots that leave environmental factor(s) as off-axis variables set to their midpoints and then explore the limits of other design factors to see if our property goals can be met. Next, you may want to investigate the range of environmental factor(s) either as on-axis or off-axis variable(s). Finally, you may choose to relax this assumption and explore optimizing with regard to all factors (including environmental ones) , to see if this makes a difference in meeting process goals albeit recognizing greater control of such factors(s) may be required.
Identify recommended factor settings (where predictions satisfy stated goals) to test model predictions with confirmatory runs. Obtaining independent data for validation is a critical step for building confidence in the predictive capabilities of our models in the region of interest. Wait for instructions on how to collect and process the confirmatory runs.
Finally, answer all thought questions on slide 12 and fill in your results on the team report spreadsheets with recommended settings, predicted response levels, and results obtained from confirmatory runs. Also, be prepared to discuss (or assign a spokesperson from your team to discuss) your teams approach, results, and recommendations.
DuPont Quality Management and Technology SOE/MTB - 15.6© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Perform initial design size or sensitivity calculations Design the experiment Run the experiment on the process and enter the data Analyze the experimental data
– fit the model to the responses– determine the important effects and check model adequacy (examine the
various tabular summaries and residuals; any evidence of LOF?)– model look OK?
Generate plots describing the model– 2D and 3D contour plots– do any predictions from the 2D contour plots satisfy process goals?– optimize the individual responses and then simultaneously optimize to meet
the combined process goals Verify predictions with check point runs
Flowchart for Experimental Design and Analysis
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Thought Questions for Workshop 3(Before contour plotting)
Do any of your initial design points satisfy the property and process goals?
Which model terms are statistically significant? What confidence level did you use to judge significance?
How well does the model fit the data? Be prepared to justify your answer.
Is there evidence of lack of fit? At a given set of experimental conditions (X settings),
what is the experimental error in BR and AS (in terms of a standard deviation)?
Based on your results, which factors or set of factors would you choose as on-axis factors in contour plots?
DuPont Quality Management and Technology SOE/MTB - 15.8© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 - Response Surface Model Results (BR)
Team
1
2
3
4
5
6
7
8
9
10
ResidsOK?
DesignUsed
Significant Effects (list factor abbreviations)Main Effects Inter. Effects Quad. Effects
LOF?R2Adj
ResdSD
RepSD
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Team
11
12
13
14
15
16
17
18
19
20
ResidsOK?
DesignUsed
Significant Effects (list factor abbreviations)Main Effects Inter. Effects Quad. Effects
LOF?R2Adj
ResdSD
RepSD
Workshop 3 - Response Surface Model Results (BR)
DuPont Quality Management and Technology SOE/MTB - 15.10© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Team
1
2
3
4
5
6
7
8
9
10
ResidsOK?
DesignUsed
Significant Effects (list factor abbreviations)Main Effects Inter. Effects Quad. Effects
LOF?R2Adj
ResdSD
RepSD
Workshop 3 - Response Surface Model Results (AS)
DuPont Quality Management and Technology SOE/MTB - 15.11© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Team
11
12
13
14
15
16
17
18
19
20
ResidsOK?
DesignUsed
Significant Effects (list factor abbreviations)Main Effects Inter. Effects Quad. Effects
LOF?R2Adj
ResdSD
RepSD
Workshop 3 - Response Surface Model Results (AS)
DuPont Quality Management and Technology SOE/MTB - 15.12© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Thought Questions for Workshop 3(After contour plotting)
Which contour plot(s) were the most informative? What is the most promising region of the design space for
satisfying the property goals? Are throughput levels OK? What confirmatory (check point) runs did you make?
Were your predictions supported? Requirements met? What conclusions can you make, if any, on the impact the
environmental factor(s) may have on meeting your process goals?
What recommendations would you make, if any, to improve the process by exercising greater control of the environmental factor(s)?
What plot(s) and / or tables would you include in a report of your results?
DuPont Quality Management and Technology SOE/MTB - 15.13© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 (Glyxel Response Surface) - Team Reports
Recommended Settings (use MP for MidPoint of excluded factors)
Team TP AC AI CA RP DT ET QT QF PT ST
1
2
3
4
5
6
7
8
9
10
PredictedBR AS
ObtainedBR AS
DuPont Quality Management and Technology SOE/MTB - 15.14© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Workshop 3 (Glyxel Response Surface) - Team Reports
11
12
13
14
15
16
17
18
19
20
Recommended Settings (use MP for MidPoint of excluded factors)
Team TP AC AI CA RP DT ET QT QF PT STPredictedBR AS
ObtainedBR AS
DuPont Quality Management and Technology SOE/MTB - 16.1© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
Section 16
OTHER EXPERIMENTALENVIRONMENTS
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SOME OTHER TYPES OFEXPERIMENTAL SITUATIONS
Categorical factors with more than 2 levels Constrained regions Mixture problems Incomplete block designs Split plot designs Nested designs Supersaturated designs
FOR THESE TYPES OF SITUATIONSCONSULT AN EXPERT!
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CATEGORICAL FACTORSWITH MULTIPLE LEVELS
Catalysts
Electronic components
Suppliers
Operators
Machines
Brands or types of formulation ingredients
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MIXTURE EXAMPLES
Drugs
Gasoline Blends
Metal Alloys
Rocket Propellants
Aerosol Propellants
Herbicides
Paints
Dyes
Textile Fiber Blends
Concrete
Cake Mixes
Composite Materials
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A CLASSIC MIXTURE
5 Parts Gin 1 Part Vermouth
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MIXTURE CONSTRAINT
Mixture Components Cannot Be Varied Independently
Factorial and Response Surface Designs Cannot Be Used
Σj=1
q
Xj = 10 ≤ Xj ≤ 1
Σso Xq = 1 - Xjj=1
q-1
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FACTOR SPACE IN TWO VARIABLES
FACTOR SPACE IN THREE VARIABLESX1
X2
Independent
X1
X2
Mixture
- + 0 1
1
0
X1
X2
X3
X1
X2
X3
Independent Mixture
X1+X2+X3=1
+
-
DuPont Quality Management and Technology SOE/MTB - 16.8© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
FACTOR SPACE FOR A FOUR-COMPONENT MIXTURE
0,1,0,0
0,0,1,0 1,0,0,0
0,0,0,1
DuPont Quality Management and Technology SOE/MTB - 16.9© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
QUADRATIC RESPONSE-SURFACEMODEL FOR TWO FACTORS
Y = a0 + a1X1 + a2X2 + a12X1X2 + a11X12 + a22X2
2
The Mixture ConstraintX1 + X2 = 1X1
2 = X1*X1 = X1(1 - X2) = X1 - X1*X2
X22 = X2*X2 = X2(1 - X1) = X2 - X1*X2
Quadratic Mixture Model (Scheffé)Y = b1X1 + b2X2 + b12X1X2
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TWO-COMPONENT MIXTUREScheffé Linear & Quadratic Model
d = 0.25*b12
Y
b1
b2
X1 1.0 0.5 0.0
X2 0.0 0.5 1.0
DuPont Quality Management and Technology SOE/MTB - 16.11© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
THREE-COMPONENT MIXTURE DESIGN
X1 = 1
X3 = 1X2 = 1
Pure Component
Binary Blend
Ternary Blend
Check Points
X1 = 0
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ROCKET PROPELLANT CONTOUR PLOT
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
*
* *
* *
*
*
* *
*
800900
X2
X3
700
1000
1000
800
Maximum Near(0.2, 0.3, 0.5)
DuPont Quality Management and Technology SOE/MTB - 16.13© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
MINIMUM COMPONENT LEVELS
Concrete
Cake
Steel
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MINIMUM COMPONENT LEVELS
0.0 0.50.0
0.5
0.0
0.5
X2 = 1
X1 = 1
X3 = 1
X1 = 0.18
X3 = 0.11
X2 = 0.14Requirements:
X1 ≥ 0.18X2 ≥ 0.14X3 ≥ 0.11
DuPont Quality Management and Technology SOE/MTB - 16.15© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
0.0 0.50.0
0.5
0.0
0.5
X3 = 1
X1 = 0.51
X3 = 0.58
X2 = 0.52
X3 = 0.11
X2 = 0.14
X1 = 0.18
X2 = 1
X1 = 1
MINIMUM AND MAXIMUM COMPONENT LEVELS
Requirements:0.18 ≤ X1 ≤ 0.510.14 ≤ X2 ≤ 0.520.11 ≤ X3 ≤ 0.58
DuPont Quality Management and Technology SOE/MTB - 16.16© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
FLARE EXPERIMENT: DESIGN
X2 = NaNO3
X3 = SrNO3 X1 = Mg
X4 = Binder
Face Centers
Vertices
Center
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AEROSOL PROPELLANT STUDY
X2= 1
X1 = 1
X3 = 1
MixtureHighly
Flammable
DuPont Quality Management and Technology SOE/MTB - 16.18© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
EXAMPLE OF CONSTRAINED REGION:Petroleum Fractionation Process
Amountof TolueneIn Solvent
Solvent/Solute
Regionof Interest
Equipment Fouling:No Phase Separation
UnfavorableEconomics
DuPont Quality Management and Technology SOE/MTB - 16.19© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
STEPS FOR DESIGNS IN IRREGULAR REGIONS
Define Region
Incorporate the Principles of Good Design
Identify Candidate Runs– Include extreme points
Select Runs– By inspection, if geometry simple– Using computer-aided algorithmic design
(e.g. D-Optimal Design)
DuPont Quality Management and Technology SOE/MTB - 16.20© 2000 E. I. du Pont de Nemours and Company Revised 7/02/2000
OTHER EXPERIMENTAL ENVIRONMENTS:Summary
Discrete Factors
Mixture Designs
Constrained Factor Spaces
DuPont Quality Management and Technology SOE/MTB - 18.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
Section 18
SUMMARY
DuPont Quality Management and Technology SOE/MTB - 18.2© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
EXPERIMENTAL DESIGN VERSUSONE-FACTOR-AT-A-TIME EXPERIMENTS
Drawbacks of one-factor-at-a-time experiments– Not space-filling
– Ignores interactions
– Ignores experimental error
– Inefficient due to lack of hidden replication
– Limited system understanding
– Potential bias error due to lack of randomization
DuPont Quality Management and Technology SOE/MTB - 18.3© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
EXPERIMENTAL DESIGN VERSUSANALYSIS OF HISTORICAL DATA
Drawbacks of historical data analysis– Correlations between factors– Unrecorded control actions may create misleading
effects, confusion of cause and effect– Typical lack of boldness in factor settings– Data collection problems
– Missing data– Bad observations
– Large bias errors due to lack of randomization– At best describes “What is” instead of “What is
possible”
DuPont Quality Management and Technology SOE/MTB - 18.4© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHEN SHOULD I USE DOE ?
Discovery Research and Scouting Product/Process Design and Development Process Scale-up, Startup, and
Qualification Process Control and Calibration Product/Process Improvement
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PRINCIPLES OF GOOD EXPERIMENTAL STRATEGY
Diagnosis of the Environment (objectives, prior knowledge, number & nature of factors)
Balanced Statistical Designs Measure All Relevant Responses Bite-Sized Experiments Boldness Randomization and Blocking Estimate Experimental Error Avoid Blunders Plan Ahead for Statistical Analysis
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EXPERIMENTAL EVOLUTION
Screening DesignsMany factorsDistinguish “critical few”
from “trivial many”Linear model: Y = b0 + b1X1 + b2X2
+ b3X3 + b4X4 + . . .Plackett-Burman and
Fractional FactorialUsually only a few moreruns than factors
Interaction DesignsFewer factorsIdentify/exploit interactionsModel contains linear terms
and at least some interactions:Y = b0 + b1X1 + b2X2
+ b3X3 + b12X1*X2+ b13X1*X3 + b23X2*X3
Full and Fractional Factorial
Response Surface DesignsSmall number of factors (3-6)Used for prediction, optimization,modelling, . . .
Quadratic model:Y = b0 + b1X1 + b2X2
+ b3X3 + b12X1*X2+ b13X1*X3 + b23X2*X3+ b11X1
2 + b22X22 + b33X3
2
Face-centered cube, Box-Behnken,others
DuPont Quality Management and Technology SOE/MTB - 18.7© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
DOE APPLICATION PROCESSStrategy of Experimentation
Gather Information
Define Experimental Objectives
Design the Experiment
Run Experiment
Analyze Experiment
Interpret Results
Perform Confirmation Runs
Go to Next Stage of Experimentation?
Apply Results
IdentifyBusinessNeeds
Assess, Document & Communicate
Business Results
UpdateInformation
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ANALYZING EXPERIMENTS: Flowchart
Rank Effects ExploitInteractions
Find “ Optimum”
ScreeningDesigns
InteractionDesigns
Response Sur faceDesigns
Main Effects Main Effectsand Interactions
QuadraticModel
ENTER DATA
DISPLAY DATA
PLOT/VERIFY DATA
FIT MODEL
ASSESS SIGNIFICANCE
VALIDATE ANALYSIS
PLOT RESULTS
DuPont Quality Management and Technology SOE/MTB - 18.9© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000
WHEN DO I NEED A D-OPTIMAL DESIGN ?
You may need a d-optimal design for the following situations:
– Discrete/Qualitative factors at more than 2 levels– Constrained regions (including mixtures)– A special model (mixed number of levels of factors or models
with some terms excluded due to your assuming that they have negligible effects on the responses of interest)
– To augment an existing design to be able to estimate a larger model (in some cases) -- assuming no change in process other than possibly a level shift (use blocking) between when the existing and new data are collected
Do not overuse d-optimal designs. Use standard designs whenever they are appropriate.
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SOE/MTB 19.1Revised 07/02/2000
Section 19
MIXTURES IN MINITAB
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SOE/MTB 19.2Revised 07/02/2000
TYPES OF MIXTURE DESIGNSAVAILABLE IN MINITAB
Simplex centroid– use when components have no upper bounds (lower bounds are OK)– includes:
• 2q-1 runs for q components• all pure component (100% of component) runs, all binary (½, ½) blends,
all ternary ( , , ) blends, … Simplex lattice
– use when components have no upper bounds (lower bounds are OK)– includes:
• degree 1 design: pure component runs• degree 2 design: pure component runs, binary blends• degree 3 design: pure component runs, binary blends, ternary runs, all
( , ) blends Extreme vertices
– use when components have upper bounds
1 31 31 3
1 3 2 3
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CREATING MIXTURE DESIGNSIN MINITAB
Start a new Minitab project From the Stat pull-down menu, select
DOEMixtureCreate Mixture Design
Select the number of components Select the type of mixture design Click on Designs… to select the points to be
included
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EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:
Choose type of design
Effects
X1X2X3
Ranges
.18 to .51
.14 to .52
.11 to .58
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Extreme Vertices Design Options:Vertices and Center Point only
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Extreme Vertices Design Options:Vertices, Center Point, Axial Points
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SOE/MTB 19.7Revised 07/02/2000
Extreme Vertices Design Options:Vertices, Center Point, Binary Blends
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SOE/MTB 19.8Revised 07/02/2000
EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:
Defining factors in Minitab
Effects
X1X2X3
Ranges
.18 to .51
.14 to .52
.11 to .58
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SOE/MTB 19.9Revised 07/02/2000
EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:
Minitab worksheet from extreme vertices design with degree=2 and center point
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SOE/MTB 19.10Revised 07/02/2000
VISUALIZING THE MIXTURE DESIGN IN MINITAB
After producing the design: Select from the Stat pull-down menu
DOEMixtureSimplex Design Plot
Click OK to usethe default graphsettings
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ANALYSIS OF MIXTURE DATAIN MINITAB
Select from the Stat pull-down menuDOEMixtureAnalyze Mixture Design
Select the response column to analyze The default model is quadratic -- click OK
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SOE/MTB 19.12Revised 07/02/2000
EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:
Minitab analysis resultsRegression for Mixtures: Response versus X1, X2, X3Estimated Regression Coefficients for Response (component proportions)
Term Coef SE Coef T P VIFX1 44.14 24.42 * * 243.89X2 4.60 20.66 * * 168.54X3 -31.81 13.44 * * 82.35X1*X2 194.74 77.83 2.50 0.041 280.33X1*X3 209.88 62.86 3.34 0.012 160.19X2*X3 165.30 49.35 3.35 0.012 77.67
S = 2.0298 PRESS = 85.656R-Sq = 98.26% R-Sq(pred) = 94.85% R-Sq(adj) = 97.02%
Analysis of Variance for Response (component proportions)
Source DF Seq SS Adj SS Adj MS F PRegression 5 1632.89 1632.8886 326.5777 79.26 0.000
Linear 2 1561.86 79.9464 39.9732 9.70 0.010Quadratic 3 71.03 71.0328 23.6776 5.75 0.027
Residual Error 7 28.84 28.8406 4.1201Total 12 1661.73
Unusual Observations for Response
Observation Response Fit SE Fit Residual St Resid1 46.000 49.697 1.321 -3.697 -2.40R
R denotes an observation with a large standardized residual
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SOE/MTB 19.13Revised 07/02/2000
EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:Minitab Response Trace Plot
Minitab’s response trace plot is similar to an effects plot for mixture components.
From the Stat pull-down menu, selectDOEMixtureResponse Trace Plot
Click OK for default settings
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SOE/MTB 19.14Revised 07/02/2000
Produce a contour plot of the model fit by selecting from the Stat pull-down menu:DOEMixtureContour/Surface (Wireframe) plots
Click on Contour plotSetupOK (for default settings)
3-D plots and optimizations are available also
EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:
Minitab contour plot
Note that this plot was produced by requesting specific contour levels.
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SOE/MTB 19.15Revised 07/02/2000
THOUGHT QUESTION
What if sample sizes test tells you only need 8 design points -- and each experiment will be very expensive.
What would you do?
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SOE/MTB 19.16Revised 07/02/2000
MIXTURE EXAMPLE 2
Create a Mixture Design with the FollowingConstraints
– Variable 1 must be less than 0.60.– Variable 2 must be less than 0.70.– Variable 3 must be GREATER than 0.20.
You will need an extreme vertices design due to the maximums given for variables 1 and 2.
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SOE/MTB 19.17Revised 07/02/2000
MIXTURE EXAMPLE 2:2 Design Alternatives
Minitab’s defaultdesign with axial
points
Minitab’s degree=2design without axial
points
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SOE/MTB 19.18Revised 07/02/2000
MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS
Four factors with their ranges.Poly 0.10 to 0.30Comp1 0.00 to 0.15Comp2 0.00 to 0.15Filler 0.55 to 0.85
We have the following additional constraintsB + C <= 0.15B + C >= 0.05.
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SOE/MTB 19.19Revised 07/02/2000
MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Define Variables
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SOE/MTB 19.20Revised 07/02/2000
MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Define Constraints
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SOE/MTB 19.21Revised 07/02/2000
MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Extreme Vertices
with Axial PointsRun Poly Comp1 Comp2 Filler
1 0.15 0.050 0.025 0.775
2 0.25 0.050 0.025 0.675
3 0.25 0.100 0.025 0.625
4 0.15 0.025 0.050 0.775
5 0.10 0.000 0.050 0.850
6 0.25 0.025 0.100 0.625
7 0.20 0.050 0.050 0.700
8 0.10 0.050 0.000 0.850
9 0.30 0.000 0.050 0.650
10 0.10 0.000 0.150 0.750
11 0.30 0.050 0.000 0.650
12 0.15 0.025 0.100 0.725
13 0.15 0.100 0.025 0.725
14 0.10 0.150 0.000 0.750
15 0.30 0.000 0.150 0.550
16 0.25 0.025 0.050 0.675
17 0.30 0.150 0.000 0.550
DuPont Quality Management and Technology SOE/MTB - 20.1© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
Section 20
GLOSSARY OF TERMS
DuPont Quality Management and Technology SOE/MTB - 20.2© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
GLOSSARY OF TERMS
algorithmic designsDesigns that are computer-generated for very specific situations, often using an algorithmtied to some particular optimality criterion, such as D-optimality.
analysis of variance (ANOVA)Procedure for analyzing data which involves partitioning the total variation into portionsexplainable by a model and unexplainable. Through appropriate partitioning, the statisticalsignificance of particular model terms or groups of terms can be tested using F-ratios.
axial pointsPoints in a central composite design having the property that all but one of the factors areset at their middle level. Syn: star points.
balanceDesirable characteristic of an experimental design wherein design points are allocated ina manner that is balanced with respect to the center of the factor space.
boldnessThe recommended practice in experimentation of investigating factors over wide ranges.
bias errorVariability in the response data that is of a systematic, patterned nature, often due to asingle assignable cause. Syn: systematic error.
blockingRunning the experiment in specially chosen subgroups, called blocks, within which theexperimental conditions or material is expected to be more homogeneous than betweenblocks.
Box-Behnken designsA particular class of response surface designs which are spherical in shape.
categorical factorA factor that can assume only a finite number of possible values or levels. The levels maybe numerical or non-numerical labels, and are usually treated as unordered. Syn:discrete factor.
centeringThe procedure of re-expressing the scale of a continuous factor by subtracting a centralvalue, such as the midpoint of the factor’s experimental range, from the factor values.Fitting polynomial models in the centered factors results in more easily-interpretedcoefficients, and reduces correlations among the coefficients.
central composite designA class of response surface designs which consist of corner points, axial points, and theoverall centroid.
central composite in cube designThe same as a face-centered cube. These are a special class of central compositedesigns in which the axial points are located in the centers of the faces of the cube.
central composite in sphere designA special class of central composite designs in which the axial points are located outsidethe faces of the cube, on the surface of the circumscribed sphere.
confidence intervalAn interval within which an unknown population parameter is estimated to lie. Theconfidence level (e.g. 95%) associated with the interval represents the long-runpercentage of times that the interval will actually include the population parameter.
DuPont Quality Management and Technology SOE/MTB - 20.3© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
GLOSSARY OF TERMSconfounding
A property of a design wherein the estimates of certain effects are correlated with eachother. When two effects are totally confounded they are inseparable from each other --they are both estimated with the same contrast. Syn: aliasing.
continuous factorA factor which can take on any value over some numerical range.
contour plotA two-dimensional plot of the relationship between two continuous factors and aresponse, in which the plot axes represent the two factors, and points of constantresponse are connected by curves, called contour lines.
correlationA measure of the degree to which values of one variable change in concert with values ofanother variable.
D-optimal designA design that minimizes the volume of the region of uncertainty of the unknown modelparameters. Usually generated algorithmically.
degrees of freedom (d.f.)The number of independent pieces of information used to fit a model (model d.f.) orestimate experimental error (residual or replicate d.f.)
designThe set of specific factor combinations to be run in the experiment. Usually specified in adesign table with factors assigned to columns and runs to rows which indicate the factorcombinations to be run.
design of experiments (DOE)A strategic process, with supporting methods and tools, for guiding the planning,execution, analysis, and application of results of experimental or developmentalprograms.
discrete factorsee categorical factor
duplicateA repeated run that does not repeat all elements of the 'run' process. Examples: makingone piece of product and measuring it more than once, or setting conditions once andmaking multiple pieces of product and measuring each. Note distinction from replicate.
effectA difference of averages: high level average - low level average. The expected change inthe response as you go from the low to high level of the factor/interaction/etc.
efficiencyA comparison of the current design vs. a theoretical optimum. May be based on any ofseveral commonly-used criteria (e.g. D-efficiency or G-efficiency). Usually expressed as apercentage between 0 and 100; 100% efficiency may not always be achievable.
environmental factorFactor that may affect product functionality but is not controlled during normal productionor use. Syn: noise factor.
experimental errorLack of repeatability (variability) in the experimental outcomes.
extrapolationMaking predictions outside the range covered by the current data.
F-testA ratio of variances or mean squares used to compare means, or to compare variances,or to test the significance of terms or groups of terms in models.
DuPont Quality Management and Technology SOE/MTB - 20.4© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
GLOSSARY OF TERMSface centered cube design
See central composite in cube design.factor
A variable which is deliberately manipulated in an experiment. Syn: independentvariable, knob, predictor, input variable, controlled variable.
factorial designExperimental design that is generated by using all possible combinations of each of thelevels of the factors; most commonly used in cases where the factors each have twolevels.
foldover designA design obtained by adding the reflection of a design to the original design (therebydoubling the number of runs); the reflection is obtained by reversing all ‘+’ and ‘-’signs in the coded design table. A foldover Plackett-Burman design will isolate the maineffects from two-way interactions. Syn: reflected design.
fractional factorial designSubset of a full-factorial design that is formed by totally confounding factor effects withcertain high-order interactions.
Gaussian distributionSee normal distribution.
hidden replicationThe apparent repeating of factor combinations when a balanced design is collapsed overvariables not involved in the effect of interest.
high-order interactionsInteraction terms that involve several (typically 3 or more) variables simultaneously.Frequently used as a basis for blocking or determining fractional factorial designs.
historical dataData taken during the normal operation of a process where factors are not varied in adeliberate, planned manner.
influenceA numerical measure of the importance of an observation in determining the fitted model.Various commonly-used measures are available, which may depend on either the locationof the point in the design space, or the response value at that point, or both.
inoperable regionA subset of the design space in which response data can not be collected/used.
interactionA condition involving two or more factors in which the effect of one factor depends on thelevels of the other(s).
interaction modelA model which consists of main effects and 2-factor interactions (higher-order interactionsgenerally not included).
interpolationMaking predictions within the range covered by the current data.
lack-of-fit testAn analysis that compares residual variability versus replicate variability to assesswhether the model could somehow be significantly improved using the current data.
linear modelA model which includes main effects only.
meanThe average.
DuPont Quality Management and Technology SOE/MTB - 20.5© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
GLOSSARY OF TERMSmixture experiment
An experimental environment where the factors of interest are the proportions of variousingredients in a formulation.
modelA mathematical representation of the relationship between the factors and the response.
normal distributionA symmetric, bell-shaped function that represents the expected frequency of data values,often used to model the distribution of random error. It is specified by a mean andstandard deviation. Syn: Gaussian distribution.
one-factor-at-a-time experimentationAn experimental strategy which involves holding all factors constant except one, which isvaried across a range. This process is then repeated in turn for each factor.
orthogonal designA design in which the columns of the design matrix are uncorrelated with each other.
orthogonal coding or scalingExpressing the range of a factor on a -1 to +1 scale.
outlierAn observation that appears to be far removed from the range of variation of the otherobservations in the data set. May suggest a possible error or anomaly.
p-value of an estimated effectThe probability of observing an effect that large purely by chance i.e. when the true effectis actually zero. The smaller the p-value, the stronger the evidence of a real effect.
parameterAn unknown constant associated with the population.
Plackett-Burman designA class of two-level screening designs that exist in multiples of 4 runs. Factor effects arenot completely independent of 2-factor interactions. Designs where the number of runs isa power of 2 should be treated as fractional factorials.
pooled standard deviationA combined estimate of experimental error variability based on replicating more than oneset of experimental conditions. Assumes that the true variance is about the same for allfactor combinations in the experiment.
populationThe hypothetical set of all possible data values of a variable.
practical significanceThe degree to which an estimated effect or parameter represents something meaningfulor useful in the context of the experimental environment.
pure error standard deviationSee replicate standard deviation.
quadratic modelA continuous factor model which consists of main effects, 2-factor interactions, andcurvature terms in each factor.
qualityThe totality of features and characteristics of a product or service that bear on its ability tosatisfy stated or implied needs.
R-squaredThe proportion of the total variation of the response explained by fitting the model. Alsoknown as the coefficient of determination. Increases with the addition of model terms,regardless of their significance.
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GLOSSARY OF TERMSR-squared adjusted
An adjusted form of R-squared that takes into account the number of terms in the modelvis-a-vis the total number of observations. Provides a more equitable measure forcomparing models of different sizes than the unadjusted R-squared. Can be negative formodels that have no value.
random errorVariability in the response data which exhibits no systematic pattern. It can not beattributable to any single cause.
randomizationThe deliberate scrambling of the run order of the design so that any bias present isunlikely to be confounded with any factor effects, but will appear as random variation.
reflected designSee foldover design.
regression (least squares)A method of fitting a model to a set of data by minimizing the sum of squares of thedeviations from the model.
regression F-testA test that assesses the overall significance of the model. Compares the varianceexplained by the model to the variance unexplained by the model.
replicateA repeated run that includes repetition of all components of the 'run' process (i.e.changingfactor settings, making product, measuring product, etc.) Note distinction from duplicate.
replicate standard deviationA pooled standard deviation computed using all replicated sets of runs. Provides anestimate of the standard deviation of experimental error. Syn: pure error standarddeviation.
residualThe difference between an observed response value and the response predicted from themodel.
residual standard deviationA standard deviation based on the set of all residual values. Under the assumption thatthe fitted model form is valid, provides an estimate of the standard deviation ofexperimental error.
resolution (of fractional-factorial design)A number which indicates the level of confounding in a fractional-factorial design. Highernumbers imply less confounding. A design of resolution R is one in which no p-factoreffect is confounded with any other effect containing fewer than R-p factors.
responseA variable which is observed/measured whose value may depend upon the settings of thedesign factors. Syn: dependent variable, property, characteristic.
response surface experimentA stage of experimentation where the experimental data is to be used for optimization,calibration, prediction, etc. Quadratic models are typically used as the fitted models.
robustInsensitive to changes in environmental conditions.
screening experimentA stage of experimentation where the experimenter needs relatively crude informationabout which of a relatively large number of factors are important. Models containing maineffects only are typically used here.
DuPont Quality Management and Technology SOE/MTB - 20.7© 2000 E. I. du Pont de Nemours and Company Revised 5/08/97
GLOSSARY OF TERMS
signal-to-noise ratioDelta/s where delta is the minimum change in the response that is desired to be detectedand s is the standard deviation of experimental error.
standard deviationA measure of the variability in a population or set of data. The square root of thevariance.
standard errorThe variability associated with an estimated effect or coefficient.
standardized residualThe residual divided by the estimated standard deviation of the residuals where allobservations contribute to the standard deviation.
statisticA numerical characteristic of a sample; for example, mean and standard deviation.
statistical significanceA conclusion made from statistical analysis of data that a difference or effect is real.
studentized residualThe residual divided by the estimated standard deviation of the residual where the currentobservation is omitted from the standard deviation calculation.
transformationA function applied to a variable (typically a response) to help improve the fit of the model,or to make the statistical analysis more valid.
variableA measurement or observation for which any of various possible data values can occur.
varianceA measure of the variability in a population or set of data. The square of the standarddeviation.
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SOE/MTB 21.1Revised: 5/08/2000
Section 21
REFERENCE
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 21.2Revised: 5/08/2000
CONTENTS
Catalogue of Designs Blank Cube Diagrams Defining Relations for Fractional-Factorial Designs Miscellaneous Formulas Selected DOE Bibliography DOE Related Accession Reports Consultant List (hand-out)
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DESIGN CATALOGUE: Two-level Designs
DesignNumber ofDistinct Points
Confoundingof Model Terms
Full Factorial 2k none
Fractional Factorial 2k-m either total or none - see pageslater in this section
Plackett-Burman multiples of 4 main effects partially that are not powers of 2 confounded w/interactions
Plackett-Burman multiples of 8 main effects clear of 2-factorplus Reflection that are not powers of 2 interactions
k = number of factors in designm = degree of fractionation
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SOE/MTB 21.4Revised: 5/08/2000
DESIGN CATALOGUE:Response Surface Designs
DesignFace-Centered Cube 3 Cubical
Box-Behnken 3 Spherical
Spherical Central- 5 SphericalComposites
FactorLevels
Shape of Design Space
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23 FACTORIAL DESIGN
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TWO 23 FACTORIAL DESIGNS
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33 FACTORIAL DESIGN
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TWO 33 FACTORIAL DESIGNS
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THREE-FACTOR FACE-CENTERED CUBE
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TWO THREE-FACTOR FACE-CENTERED CUBES
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THREE-FACTOR BOX-BEHNKEN DESIGN
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SOE/MTB 21.12Revised: 5/08/2000
TWO THREE-FACTOR BOX-BEHNKEN DESIGNS
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FRACTIONAL-FACTORIAL DESIGNSAVAILABLE WITH VARIOUS RESOLUTIONS
Number of Resolution Resolution Resolution FullFactors III IV V or more Factorial
3
4
5
6
7
8
9
10
11
12
4
-
8
8
8
-
16
16
16
16
-
8
-
16
16
16
32
32
32
32
-
-
16
32
64
64
128
128
128
256
Number of runs required for:
8
16
32
64
128
256
512
1024
2048
4096
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SOE/MTB 21.14Revised: 5/08/2000
Standard Error of a Factor Effect
√ n1 n2where n1 and n2 are the number of observations in the low and high “halves” of the factor effect.
Confidence Intervalestimate +/- t * standard errort is a tabled Student-t quantile whose value depends
on the degrees of freedom & confidence usedWhen zero is not included in the confidence interval, the effect is
statistically significant
Experiment SizeFor 2-level designs
7 or 8 2 where ∆ is effect size desired to
∆ / s detect and s = estimate of std dev.For 3-level designs
about 50% more than for 2 levels
Sample MeanY1 + Y2 + . . . Yn Yi
n nSample Variance
(Y1-Y)2 + (Y2-Y)2 + . . . + (Yn-Y)2
n - 1Sample Pooled Variance
(n1-1)s12+(n2-1)s2
2+ . . .+(nk-1)sk2
(n1-1)+(n2-1)+ . . .+(nk-1)where s1
2 , s22 , . . . sk
2 are the individual variancesand n1 , n2 , . . . nk are the number of replicate measurements at each combination
Sample Standard Deviationss = √ s2 sp = √ sp
2
Estimated EffectYhigh - Ylow
Y =
)(n =
MISCELLANEOUS HANDY FORMULAS
1 1+Σi=1
n
=
s2 =
sp2 =
sFE = sp
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 21.15Revised: 5/08/2000
Explained Variation (%) in Regression (R-Squared)
Explained Variation in Regression (R-Squared adjusted)
MISCELLANEOUS HANDY FORMULAS(continued)
Σi=1
n
(Yi - Y)2_
(Ypredicted - Y)2Σi=1
n
Σi=1
n
(Yi - Y)2
(Yi - Ypred)2Σi=1
n
_
_
R2(adj) = 100 1 -
where k = number of model terms (including constant)
R2(adj)= 100 (1 - (varianceresidual/ variancetotal))
)(
100
/(n-k)
/(n-1)
R2 =
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 21.16Revised: 5/08/2000
SELECTED DOE BIBLIOGRAPHY Design and Analysis of Experiments, Douglas C. Montgomery, 3rd ed.,
1990, John Wiley and Sons, Inc. Empirical Model Building and Response Surfaces, George E. P. Box
and Norman R. Draper, 1987, John Wiley and Sons, Inc. Experiments with Mixtures, John Cornell, 2nd ed., 1990, John Wiley and
Sons, Inc. Response Surface Methodology, Raymond H. Myers, 1976, Virginia
Polytechnic Institute and State University. Statistical Design & Analysis of Experiments With Applications to
Engineering and Science, Robert L. Mason, Richard F. Gunst, James L. Hess, 1989, John Wiley & Sons, Inc.
Statistics for Experimenters, George E. P. Box, William G. Hunter and J. Stuart Hunter, 1978, John Wiley and Sons, Inc.
Strategy of Experimentation, Course Text (1988), DuPont Quality Management & Technology Center, Wilmington, DE.
DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company
SOE/MTB 21.17Revised: 5/08/2000
AVAILABLE DOE-RELATED MATERIALS Design of Experiments: A Competitive Advantage
– Introduction in question and answer format– Accession Report #17934
Design of Experiments Application Guide– Outline of DOE as a strategic process. How and when to apply the
tools and methods effectively.– Accession Report #17960
Design of Experiments Quick Reference Guide– Quick reference guide for trained users of DOE as a strategic
process. Includes tools and methods, examples and catalogs.– Accession Report #17961
Design of Experiments Overview (presentation)
DuPont Quality Management and Technology SOE/MTB© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000
TABLE OF CONTENTS
SECTION TITLE
1 Introduction2 Workshop 13 Foundations of the Strategy4 Factorial Geometry5 Factorial Example: Design6 Analysis of Two-Level Factorial Designs7 Factorial Example: Analysis8 Good Experimental Practice9 Screening Designs10 Screening Example11 Workshop 2 - Glyxel Screening12 Response Surface Designs13 Model Diagnostics14 Response Surface Example15 Workshop 3 - Glyxel Response Surface16 Other Experimental Environments17 Algorithmic Design18 Summary19 Mixtures in Minitab20 Glossary of Terms21 Reference
Agenda Additions:
Questions & AnswersYour Data Session